pythagoras problem solving worksheet

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Pythagorean Theorem Worksheets

These printable worksheets have exercises on finding the leg and hypotenuse of a right triangle using the Pythagorean theorem. Pythagorean triple charts with exercises are provided here. Word problems on real time application are available. Moreover, descriptive charts on the application of the theorem in different shapes are included. These handouts are ideal for 7th grade, 8th grade, and high school students. Kick into gear with our free Pythagorean theorem worksheets!

Identifying Right Triangles

Apply Pythagorean theorem to identify whether the given triangle is a right triangle. Each printable worksheet consists of six problems.

• Download the set

Pythagorean Theorem Chart

These descriptive charts explain the Pythagorean theorem with an illustration. These pdfs emphasize the relation of the theorem derived as an equation.

Pythagorean Triple

A set of three numbers is given in each problem. Grade 7 and grade 8 students need to apply the theorem and identify whether the set of numbers forms a Pythagorean triple.

Pythagorean Triple Chart

This section comprises of Pythagorean triple sets up to 100. Besides, Pythagorean triple formulas with examples are provided in the charts.

Unknown Side of a Right Triangle

Apply Pythagorean theorem to find the unknown side of the right triangle. Round the answer to the nearest tenth.

Unknown Length: Shapes

Apply the Pythagorean theorem to find the unknown length of each shape in these printable worksheets. Round the answer to the nearest tenth.

Word Problems | Level 1

Brighten your math class with this bundle of real-life word problems based on the Pythagorean Theorem. Solve each word problem by finding the missing hypotenuse of the right triangle and rounding off the answer to the nearest tenth.

Word Problems | Level 2

Presenting word problems with clear illustrations, these pdf worksheets require high school students to plug in the known values into the equation form of the Pythagorean Theorem and figure out the unknown side of the right triangle.

Related Worksheets

» Pythagorean Inequality Theorem

» Triangle

» Triangle Inequality Theorem

» Trigonometry

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Pythagorean Theorem Worksheet

48 pythagorean theorem worksheet with answers [word + pdf].

The simplicity of the Pythagorean Theorem worksheet is the best thing about it. What is the Pythagorean Theorem? Formulated in the 6th Century BC by Greek Philosopher and mathematician Pythagoras of Samos, Pythagorean Theorem is a mathematic equation used for a variety of purposes. Over the years, many engineers and architects have used Pythagorean Theorem worksheet to complete their projects .

Table of Contents

• 1 Pythagorean Theorem Worksheet
• 2 Knowing Pythagoras of Samos and how he came up with the Pythagorean equation
• 3 Understanding Pythagorean Theorem
• 4 Pythagorean Theorem Word Problems Worksheet
• 5 Using Pythagorean Theorem worksheet
• 6 Conclusion

A simple equation, Pythagorean Theorem states that the square of the hypotenuse (the side opposite to the right angle triangle) is equal to the sum of the other two sides . Following is how the Pythagorean equation is written:

In the aforementioned equation, c is the length of the hypotenuse while the length of the other two sides of the triangle are represented by b and a. Though the knowledge of the Pythagorean Theorem predates the Greek Philosopher, Pythagoras is generally credited for bringing the equation to the fore. This is the reason the Pythagorean equation is named after him. Before we discuss the Pythagorean Theorem and the Pythagorean Theorem worksheet in detail, let’s take a look at who Pythagoras of Samos was and how he came up with the Pythagorean equation.

Knowing Pythagoras of Samos and how he came up with the Pythagorean equation

A 6 th century BC Greek philosopher and mathematician, Pythagoras of Samos is widely credited for bringing the Pythagorean equation to the fore. Though others used the relationship long before his time, Pythagoras is the first one who made the relationship between the lengths of the sides on a right-angled triangle public. This is why he’s regarded as the inventor of the Pythagorean equation.

Apart from being a philosopher and mathematician, Pythagoras founded the Pythagoreanism movement. Born in Croton, Italy, Pythagoras travelled to many different countries including Greece, Egypt, and India. After moving back to Croton in 530 BC, Pythagoras established some kind of school. He returned to Samos in 520 BC. It was in late 6 th Century BC that Pythagoras started to make important contributions to philosophy and math. The Pythagorean equation was one of those contributions.

Though he revealed the Pythagorean equation to the world in the late 6 th Century BC while living in Samos, many historians believe that Pythagoras first thought about the equation during his time in Egypt. In fact, according to many historians, Pythagoras learned geometry, the Phoenicians arithmetic and other branches of mathematics from the Egyptians.

Though he has made many important contributions to philosophy, Pythagoras is widely known as the founder of the Pythagorean Theorem. As previously mentioned, the Pythagorean Theorem is a mathematical equation that states that the square of the hypotenuse (the side opposite to the right angle triangle) is equal to the sum of the other two sides .

Today the aforementioned equation bears Pythagoras’s name but it’s important to know that he wasn’t the first one to use the equation. Before Pythagoras’s time, the Indians and the Babylonians utilized the Pythagorean Theorem or equation. Since they constructed the first proof of the theorem, Pythagoras and his disciples are regarded as the inventors of the equation.

Many historians say that Pythagoras worked in a very secretive manner. This is the reason little evidence is available that the Greek Philosopher/ mathematician himself worked on and proved the Pythagorean Theorem. It is important to note that the first time Pythagoras was given credit for the Theorem was five centuries after his death. This makes Pythagoras’s contribution to the Theorem even more debatable. Nonetheless, since Pythagoras is the only one connected to the Pythagorean Theorem known today, we have to give him due credit. Now that we’ve discussed who Pythagoras of Samos was and how he came up with the Pythagorean equation, it’s time to take a detailed look at the Pythagorean Theorem and the Pythagorean Theorem worksheet.

Understanding Pythagorean Theorem

According to Pythagorean Theorem, the sum of the squares on the right-angled triangle’s two smaller sides is equal to the side opposite to the right angle triangle (the square on hypotenuse). Using a Pythagorean Theorem worksheet is a good way to prove the aforementioned equation. An amazing discovery about triangles made over two thousand years ago, Pythagorean Theorem says that when a triangle has a 90° angle and squares are made on each of the triangle’s three sides, the size of the biggest square is equal to the size of the other two squares put together! A short equation, Pythagorean Theorem can be written in the following manner:

In Pythagorean Theorem, c is the triangle’s longest side while b and a make up the other two sides. The longest side of the triangle in the Pythagorean Theorem is referred to as the ‘hypotenuse’. Many people ask why Pythagorean Theorem is important. The answer to this is simple: you’ll be able to find the length of a right-angled triangle’s third side if you know the length of the other two sides. This equation works like magic and can be used to find any missing value. Following is an example that uses the Pythagorean Theorem to solve a triangle.

In this equation, the longest side of the triangle ‘c’ is missing. By finding out the sum of the squares of the two other sides, we were able to find the missing value. The most famous mathematical contribution of Pythagoras, the Pythagoras Theorem was one of the earliest documented theorems. Though Pythagoras is given most of the credit for the theorem, a major contribution to the theorem was made by his students.

When you look at a Pythagoras Theorem worksheet, you’ll notice that the theorem enables you to find the length of any right angle triangle side provided you know the length of the other two sides. Also, using the theorem, you can check whether a triangle is a right triangle. The Pythagoras Theorem is extremely useful in solving many math problems. Further, you can use it in many real life situations. This is illustrated by a Pythagoras Theorem worksheet.

Pythagorean Theorem Word Problems Worksheet

Using Pythagorean Theorem worksheet

A good way to review the Pythagoras Theorem and expand the mathematical equation is using a Pythagoras Theorem worksheet. By using the worksheet, you’ll be able to get a good understanding of geometry. Additionally, the worksheet will give you an opportunity to review the knowledge related to the different types of triangles. Finally and most importantly, you’ll be able to practice the ancient equation invented by the Greek mathematician and philosopher, Pythagoras. Before you start using the Pythagoras Theorem worksheet, just remember that ‘c’ is the hypotenuse while the shorter sides of the triangle are represented by ‘a’ and ‘b’.

A Pythagoras Theorem worksheet presents students with triangles of various orientations and asks them to identify the longest side of the triangle i.e. the hypotenuse. As you know by now, the formula used in Pythagoras Theorem is a²+b²=c². Regardless of what the worksheet asks the students to identify, the formula or equation of the theorem always remain the same. Though, the students could be presented with different challenges including solving triangles:

• Labeled in different order
• With a different set of letters
• By using vertices to name the sides

The symbols used in the Pythagoras Theorem are something students will find on their calculators. Figuring out how to use these functions is what students need to establish. There is involvement of the Babylonians and the Egyptians in the invention of the Pythagoras Theorem but the earliest known proof of the theorem was produced by the school of Pythagoras.

Many Pythagorean triples were known to the Babylonians while the Egyptians knew and used the (3, 4, 5) triple. The Chinese and Indians also played a role in the invention of the Pythagoras Theorem. The first diagrammatic proof of the theorem was produced by the Chinese while the Indians discovered many triples. In 1995, the theorem became part of the Guinness Book of Records as the most proved theorem of all time.

The triples used in the Pythagoras Theorem include (3, 4, 5), (6, 8, 10), (5, 12, 13), (8,15,17), (7,24,25), (20,21,29), (12,35,37), (9,40,41), (28,45,53), (11,60,61), (16,63,65), (33,56,65) and (48,55,73). The aforementioned triples aren’t multiples of a smaller triple and the name given to them is ‘primitive’ triples. To solve a particular problem, the Pythagoras Theorem can be arranged. For example, if you’re asked to find b which is one of the two smaller sides of the right-angled triangle, you can rearrange the theorem to b²=c²-a². By doing this, you’ll be able to easily find the missing value.

The Pythagoras Theorem has many different proofs. However, when checking your answers, following are the two things that you must always remember:

• The side opposite to the right angle or simply the hypotenuse is always the longest side of the triangle
• Though it is the longest side of the triangle, the size of the hypotenuse can never exceed the sum of the other two squares

To understand this better, take a look at a Pythagoras Theorem worksheet. Today, you can get easy access to Pythagorean Theorem worksheet with answers. Nonetheless, we’re going to try and understand the Pythagoras Theorem as much as we can.

As mentioned earlier, if you know the size of the other two sides, you will be able to find out the length of the third side of the right angle triangle. Also, after being squared, the shorter length is subtracted from the square of the hypotenuse when the hypotenuse is one of the two known lengths. As seen earlier, the lengths of each side of the triangle in the Pythagoras Theorem are whole numbers. Such triangles are known as Pythagorean triangles.

Though there are many different proofs of the Pythagoras Theorem, only three of them can be constructed by students and other people on their own. The first proof starts off as rectangle and is then divided into three triangles that individually contain a right angle. To see first proof, you can use a computer or something as straight forward as an index card cut up into right triangles.

Beginning with a rectangle, the second proof of the Pythagoras Theorem starts off by constructing rectangle CADE with BA=DA. This is followed by the construction of the <BAD’s angle bisector. Once constructed, the bisector is allowed to intersect ED at point F. This makes <BAF and <DAF congruent, BA=DA, and AF=AF. This in turn makes the triangle DAF equal to triangle BAF which means that since ADF is a right angle, ABF will also be a right angle. The third and final proof of the Pythagorean Theorem that we’re going to discuss is the proof that starts off with a right angle. In this proof, triangle ABC is right angle and its right side is angle C.

The three proofs stated above are just few of the many Pythagoras Theorem. You’ll come across these proofs when you take a look at the Pythagorean Theorem worksheet with answers. Learning and understanding the Pythagorean concept is extremely important for students and other people who’ll use this theorem in their practical life.

It is important that you understand the algebraic representation of the Pythagoras Theorem as well as the geometric concepts behind it. You can accomplish this by using proofs, manipulatives, and computer technology. By using these methods to learn Pythagorean Theorem, you’ll be able to see the connections and benefit greatly.

Formulated in the 6th Century BC by Pythagoras of Samos, Pythagoras Theorem is widely used today. If you want to practice Pythagoras Theorem then you can do that easily. Pythagoras Theorem worksheets with answers are easily available and you can use these worksheets to get a good grip of the Theorem.

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Pythagorean Theorem Exercises

Pythagorean theorem practice problems with answers.

There are eight (8) problems here about the Pythagorean Theorem for you to work on. When you do something a lot, you get better at it. Let’s get started!

Here’s the Pythagorean Theorem formula for your quick reference.

Note: drawings not to scale

Problem 1: Find the value of [latex]x[/latex] in the right triangle.

[latex]10[/latex]

Problem 2: Find the value of [latex]x[/latex] in the right triangle.

[latex]4\sqrt 7 [/latex]

Problem 3: Find the value of [latex]x[/latex] in the right triangle.

[latex]3[/latex]

Problem 4: The legs of a right triangle are [latex]5[/latex] and [latex]12[/latex]. What is the length of the hypotenuse?

[latex]13[/latex]

Problem 5: The leg of a right triangle is [latex]8[/latex] and its hypotenuse is [latex]17[/latex]. What is the measure of its other leg?

[latex]15[/latex]

Problem 6: Suppose the shorter leg of a right triangle is [latex]\sqrt 2[/latex]. The longer leg is twice the shorter leg. Find the hypotenuse.

[latex]\sqrt {10} [/latex]

Problem 7: The hypotenuse of a right triangle is [latex]4\sqrt 2[/latex]. If the longest leg is half the hypotenuse, what is the length of the shortest leg?

[latex]2\sqrt 6[/latex]

Problem 8: Find the value of [latex]x[/latex] of the right isosceles triangle.

[latex]\sqrt 6[/latex]

You might also like these tutorials:

• Pythagorean Theorem
• Pythagorean Triples
• Generating Pythagorean Triples

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Pythagoras Theorem Questions

Welcome to our Pythagoras' Theorem Questions area. Here you will find help, support and questions to help you master Pythagoras' Theorem and apply it.

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Pythagoras' Theorem Questions

Here you will find our support page to help you learn to use and apply Pythagoras' theorem.

Please note: Pythagoras' Theorem is also called the Pythagorean Theorem

There are a range of sheets involving finding missing sides of right triangles, testing right triangles and solving word problems using Pythagoras' theorem.

Using these sheets will help your child to:

• learn Pythagoras' right triangle theorem;
• use and apply the theorem in a range of contexts to solve problems.

Pythagoras' Theorem

Pythagoras' Theorem - in more detail

Pythagoras' theorem states that in a right triangle (or right-angled triangle) the sum of the squares of the two smaller sides of the triangle is equal to the square of the hypotenuse.

In other words, \[ a^2 + b^2 = c^2 \]

where c is the hypotenuse (the longest side) and a and b are the other sides of the right triangle.

What does this mean?

This means that for any right triangle, the orange square (which is the square made using the longest side) has the same area as the other two blue squares added together.

Other formulas that can be deduced from the Pythagorean theorem

As a result of the formula \[ a^2 + b^2 = c^2 \] we can also deduce that:

• \[ b^2 = c^2 - a^2 \]
• \[ a^2 = c^2 - b^2 \]
• \[ c = \sqrt{a^2 + b^2} \]
• \[ b = \sqrt {c^2 - a^2} \]
• \[ a = \sqrt {c^2 - b^2} \]

Pythagarean Theorem Examples

Example 1) find the length of the missing side..

In this example, we need to find the hypotenuse (longest side of a right triangle).

So using pythagoras, the sum of the two smaller squares is equal to the square of the hypotenuse.

This gives us \[ 4^2 + 6^2 = ?^2 \]

So \[ ?^2 = 16 + 36 = 52 \]

This gives us \[ ? = \sqrt {52} = 7.21 \; cm \; to \; 2 \; decimal \; places \]

Example 2) Find the length of the missing side.

In this example, we need to find the length of the base of the triangle, given the other two sides.

This gives us \[ ?^2 + 5^2 = 8^2 \]

So \[ ?^2 = 8^2 - 5^2 = 64 - 25 = 39 \]

This gives us \[ ? = \sqrt {39} = 6.25 \; cm \; to \; 2 \; decimal \; places \]

Pythagoras' Theorem Question Worksheets

The following questions involve using Pythagoras' theorem to find the missing side of a right triangle.

The first sheet involves finding the hypotenuse only.

A range of different measurement units have been used in the triangles, which are not drawn to scale.

• Pythagoras Questions Sheet 1
• PDF version
• Pythagoras Questions Sheet 2
• Pythagoras Questions Sheet 3
• Pythagoras Questions Sheet 4

Pythagoras' Theorem Questions - Testing Right Triangles

The following questions involve using Pythagoras' theorem to find out whether or not a triangle is a right triangle, (whether the triangle has a right angle).

If Pythagoras' theorem is true for the triangle, and c 2 = a 2 + b 2 then the triangle is a right triangle.

If Pythagoras' theorem is false for the triangle, and c 2 = a 2 + b 2 then the triangle is not a right triangle.

• Pythagoras Triangle Test Sheet 1
• Pythagoras Triangle Test Sheet 2

Pythagoras' Theorem Questions - Word Problems

The following questions involve using Pythagoras' theorem to solve a range of word problems involving 'real-life' type questions.

On the first sheet, only the hypotenuse needs to be found, given the measurements of the other sides.

Illustrations have been provided to support students solving these word problems.

• Pythagoras Theorem Word Problems 1
• Pythagoras Theorem Word Problems 2

Geometry Formulas

• Geometry Formula Sheet

Here you will find a support page packed with a range of geometric formula.

Included in this page are formula for:

• areas and volumes of 2d and 3d shapes
• interior angles of polygons
• angles of 2d shapes
• triangle formulas and theorems

This page will provide a useful reference for anyone needing a geometric formula.

Triangle Formulas

Here you will find a support page to help you understand some of the special features that triangles have, particularly right triangles.

Using this support page will help you to:

• understand the different types and properties of triangles;
• understand how to find the area of a triangle;
• know and use Pythagoras' Theorem.

All the free printable geometry worksheets in this section support the Elementary Math Benchmarks.

• Geometry Formulas Triangles

Here you will find a range of geometry cheat sheets to help you answer a range of geometry questions.

The sheets contain information about angles, types and properties of 2d and 3d shapes, and also common formulas associated with 2d and 3d shapes.

Included in this page are:

• images of common 2d and 3d shapes;
• properties of 2d and 3d shapes;
• formulas involving 2d shapes, such as area and perimeter, pythagoras' theorem, trigonometry laws, etc;
• formulas involving 3d shapes about volume and surface area.

Using the sheets in this section will help you understand and answer a range of geometry questions.

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Pythagorean Theorem Worksheets

Related Topics: More Math Worksheets More Printable Math Worksheets 7th Grade Math

There are five sets of triangle worksheets:

• Triangle Sum Theorem
• Exterior Angle Theorem
• Area of Triangle

Area of Triangle using Sine

• Pythagorean Theorem

Examples, solutions, videos, and worksheets to help grade 7 students learn how to use the Pythagorean Theorem.

How to use the Pythagorean Theorem?

There are five sets of Pythagorean Theorem worksheets:

• Find the length of the hypotenuse
• Find the length of a side
• Find the length in two steps
• Check for right triangle
• Pythagorean Theorem Word Problems

The Pythagorean Theorem also called the Pythagoras’ Theorem is a fundamental principle in geometry that relates the sides of a right triangle.

It states that in a right triangle (a triangle with one 90-degree angle), the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Mathematically, the Pythagorean Theorem can be expressed as: c² = a² + b²

• c represents the length of the hypotenuse.
• a and b represent the lengths of the other two sides.

The Pythagorean Theorem is widely used to calculate the length of any one side of a right triangle when the lengths of the other two sides are known. It is an essential tool in geometry and has applications in various fields, including trigonometry, engineering, physics, and architecture.

The Pythagorean Theorem can be used in various ways, including:

• Finding length of any one side of a right triangle when the lengths of the other two sides are known.
• Checking if a Triangle is Right: You can use the theorem to check if a triangle is a right triangle by verifying if the relationship a² + b² = c² holds.
• Solving Problems Involving Right Triangles: The theorem is used to solve problems involving right triangles, such as finding distances, heights, or angles in various real-world situations.

For example, if you have a right triangle with one side (a) measuring 3 units and the other side (b) measuring 4 units, you can use the Pythagorean Theorem to find the length of the hypotenuse (c):

c² = a² + b² c² = 3² + 4² c² = 9 + 16 c² = 25

To find c, take the square root of both sides:

c = √25 c = 5

So, the length of the hypotenuse (c) is 5 units. This illustrates how the Pythagorean Theorem can be used to solve for side lengths in a right triangle.

Click on the following worksheet to get a printable pdf document. Scroll down the page for more Pythagorean Theorem Worksheets .

More Pythagorean Theorem Worksheets

Printable (Answers on the second page.) Pythagorean Theorem Worksheet #1 (find hypotenuse) Pythagorean Theorem Worksheet #2 (find a side) Pythagorean Theorem Worksheet #3 (2-step) Pythagorean Theorem Worksheet #4 (check for right triangle) Pythagorean Theorem Worksheet #5 (word problems)

Online Pythagorean Theorem Find the missing side Test for right triangle Pythagorean Theorem Word Problems Converse Pythagorean Theorem, Types of Triangles

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Pythagorean Theorem Worksheet: 4 Levels

Looking for the perfect Pythagorean theorem worksheet? Need some right-angle triangle practice? Look no further!

The theorem may be short and seem simple, but there are many types of questions you could face in an exam.

You will use this theorem if you continue studying math at a higher level. It’s a useful equation to keep in your math tool kit !

Applying the theorem with confidence is helpful for many people in the world.

The Pythagorean theorem really does have lots of real-life uses. You may use it for crafts, in construction, for baking, gardening, or choosing a smartphone. Without it, GPS location wouldn’t be a thing!

Each worksheet has a different focus. You might have one particular area you’d like to improve on, or perhaps you’re looking for a complete crash course.

Either way, don’t forget to show your working out, and start at the right skill level – each Pythagorean theorem worksheet is given in order of difficulty!

Worksheets The Man, the Legend: Pythagoras of Samos Who ACTUALLY Wrote the Pythagorean Theorem? What is the Theorem Other Theories and Discoveries Attributed to Pythagoras How Many Solutions? Solutions When a = 2 Special Solutions: Triples Properties of Triples A Brief Example Families of Triples A Briefer Example To Sum Up (Pun Intended!)

Free Printable Pythagorean Theorem Worksheet With Answers

Level 1: identify the right-angle triangles.

Level 2: Identify the Pythagorean Triples

Level 3: Find the Missing Side Length of Right-Angle Triangles

Level 4: Real-World Pythagorean Theorem Word Problems (2 Pages)

The Man, the Legend: Pythagoras of Samos

The Pythagorean theorem is named after the Greek mathematician and philosopher, Pythagoras, who was born around 570 BC.

His home was the small Greek island Samos which is in the Mediterranean Sea, near Turkey.

Although he grew up in a beautiful paradise, there was political conflict. He traveled to Egypt and Persia as a young man before settling in southern Italy.

In Crotone, Italy, he established a school of philosophy and science. His students were called his disciples, his school like a family – a brotherhood.

In fact, the group he established was called The Brotherhood of Pythagoreans.

The Brotherhood had a structured life of exercise and study. They believed that to live forever in their second life , they needed to be physically perfect.

Pythagoras and his disciples, as they were called, were incredibly disciplined. They wanted to keep their bodies “morally pure”.

He was, by most accounts, a wacky guy – he always put his right shoe on first, never ate beans, didn’t travel by public roads, and encouraged his students to be silent wherever possible!

None of his writings have survived. That isn’t too surprising, given that he lived almost 2600 years ago!

Pythagoras is a man of mystery. Without his records, everything we know about him comes from what his disciples and other scholars wrote about him.

Imagine trying to find out what someone else is like, and what discoveries they have made, from their friends and family’s posts on social media. You wouldn’t get all the detail you need!

It’s known that he taught a special “Pythagorean Way of Life,” based on religion and ritual. Interestingly though, his beliefs were shaped by mathematics.

After he died in roughly 490BC, many people glorified Pythagoras and considered him God-like.

Apparently, he performed miracles: he killed a dangerous snake by biting it, he was seen in two cities at the same time, a river spoke to him and his thigh was golden.

Click here to go back to the Pythagorean theorem worksheet .

Who ACTUALLY Wrote the Pythagorean Theorem?

The Pythagorean Theorem is what Pythagoras is most famous for – it’s probably the first thing you think of when you hear his name.

But did Pythagoras himself really think of the theorem? Historians don’t know! It’s possible that one of his disciples wrote it after Pythagoras died, and named it after him.

The Pythagoreans were secretive. They came up with many proofs of important mathematical ideas but didn’t share their discoveries with anyone else.

If Pythagoras really did write the famous theorem, he didn’t tell the world about it!

What is the Pythagorean Theorem?

First, a right-angled triangle is a triangle that has an internal angle of 90°. It looks a bit like a ladder leaning between the floor and a wall.

Second, the hypotenuse of a right-angled triangle is its longest side. It is always the diagonal and never touches the right-angle. It would be the ladder in this picture!

“In a right-angled triangle, the square of the hypotenuse equals the sum of squares of the other two sides”.

Don’t let the word ‘theorem’ put you off! It boils down to one equation:

Where a and b are the smaller side lengths and c is the length of hypotenuse .

If you have two lengths of a right-angled triangle, you can find the third using the formula.

Other Theories and Discoveries Attributed to Pythagoras

He was a well-rounded guy!

In Ancient Greek society, Pythagoras was credited with being the first to teach that the Earth is a sphere.

He was also one of the first to divide the world into five climatic zones – two cold zones at the poles, a tropical zone across the equator, and milder zones between them.

Pythagoras believed that math and abstract thought are the basis of morality and science. His teachings in philosophy inspired famous philosophers Plato and Aristotle.

Pythagoras started the idea of a number system, founded on his belief that numbers make up everything in the world.

He believed that all numbers are rational – that they can be written as fractions – and that they were beautiful in their simplicity.

The Pythagoreans were shocked when they discovered \(\sqrt{2\ }\), an irrational number that cannot be written as a fraction.

Pythagoras linked math to music when he heard blacksmiths’ hammers hitting metal. He realized that the pitch of the sound made was related to the size of the hammer.

Certain pitches sound nice when played together – they are in consonance. Other pitches clash, making a nasty sound – they are in dissonance.

Playing notes with special intervals between their pitches produces harmonies. Thank Pythagoras next time you hear singers harmonizing!

How Many Solutions?

There are infinitely many solutions to the equation a 2 +b 2 =c 2

This becomes clear if one of the values is fixed. Let’s suppose that a=2 and see how many pairs of b and c values make sense.

This equation gives the relationship between b and c . Any value of c>2 has a corresponding value of b which satisfies the Pythagorean theorem because the radical is positive.

If c<2 , the equation has imaginary, or complex numbers .

For example, if we let c=1 , then

which isn’t a real number, because no real number can multiply by itself to make a negative number!

Solutions When a=2

We have just said that you can pick any value of c>2 and plug this into the equation \(b=\sqrt{c^2 – 4}\) to complete the solution.

Let’s give this a try:

If you stop to think about it, what would a right-angle triangle with two sides of the same length look like? Let’s use math to find out.

We’re going to let a=2 and c=2 , and put them into the equation to see what happens.

Hmmm, a right-angle triangle with one side of zero length? The math has spoken – there is no such thing!

There are infinitely many values of c to pick from, so there are infinitely many corresponding values of b , so there are infinitely many solutions to the Pythagorean Theorem when a=2 !

This thinking can be applied to any fixed value of a . There are infinitely many solutions for each fixed value of a , and there are infinitely many a ’s to pick from.

In short, that’s a lot of solutions!

Special Solutions: Pythagorean Triples

The possibilities are literally endless when it comes to solving this equation.

It’s also interesting to study a subset of solutions, called Pythagorean triples .

These are triplets of positive integers a , b and c which solve the equation a 2 +b 2 =c 2 .

The first – and most famous! – example is 3 2 +4 2 =5 2 which, when used for measuring, is often referred to by tradespeople as 3, 4, 5.

Pythagorean triples are easier to work with because no radicals are involved. However, there is less freedom when creating them.

These triplets aren’t constructed by picking two variables and solving for the third – because of the special integer property, we must be more sophisticated!

Properties of Pythagorean Triples

The integer triplets \((a,b,c)\) fit into one of two categories: either all three numbers are even, or one is even and the other two are odd.

This is because the squares of even numbers are even, and the squares of odd numbers are odd.

Adding to this property, some equations help us construct a valid Pythagorean triple.

Begin by picking two positive integers. Call the bigger number x , and the smaller number y . It’s important that x and y are not equal to each other.

Substituting x and y into the following equations will give your triple:

Set x=10, and y=5. This is valid because \(x\neq y\) and \(x>y\).

The Pythagorean triple is

It’s written as:

In this case, two of the numbers are odd and one is even so it’s in the second category.

We can check that the triple satisfies the Pythagorean theorem by simply substituting in a , b , and c to find that:

Families of Triples

Equations can be manipulated by doing the same thing to both sides. Even though it has a fancy name, Pythagoras’ equation is no different!

If you multiply both sides of an equation by a positive integer, it’s still valid.

This idea is super useful – multiplying a triple by a positive integer will always give another triple!

Multiplying a triple is the same as multiplying both sides of the corresponding Pythagorean equation.

Each number in the triple is multiplied by a positive integer, giving another positive integer.

Think of it as scaling up the triangle, dragging it by the corner as you would with a photo or textbox on a computer.

Dividing triples is a dangerous game – if the divisor is not a factor of each number in the triple, the resulting numbers will not be integers so a Pythagorean triple will not be made. Disaster!

Every triple can be scaled upwards but not all can be scaled downwards.

When multiplying triples, you’ve got a lot of options. Infinitely many, in fact!

This means that there is an infinite family of triples, related to one another by multiplication.

In each ‘family’ of related triples, the smallest is given a special name.

A Pythagorean triple is called primitive if the greatest common divisor of a , b and c is 1.

Primitive triples are the simplest elements in their families – they cannot be divided to produce a smaller triple.

Let’s return to the previous triple we found: (75, 100, 125) .

This triple is not primitive because of the greatest common divisor, or GCD , which is 25, not 1.

Dividing each number by 25 gives another Pythagorean triple:

The division within the exponent looks a little suspicious, but we aren’t breaking any rules.

It is the same as dividing both sides by 25 2 , which is allowed!

To Sum Up (Pun Intended!)

I hope you’ve enjoyed learning some history along with your math. Pythagoras is a legendary figure, shrouded in mystery, but his theorem doesn’t have to be.

After using a Pythagorean theorem worksheet, you should have the confidence to identify right-angled triangles, find missing side lengths, identify Pythagorean triples, solve real-life problems, and use the theorem in trickier shapes.

Each worksheet uses the same amazing little equation a 2 +b 2 =c 2 , discovered around 2500 years ago.

Pythagoras is most famous for this theorem, but he had a busy life! One of his achievements was creating a circle of scholars called The Brotherhood of Pythagoreans.

They were influential in Ancient Greek society – Pythagoras’s work in philosophy inspired Plato and Aristotle, and the Brotherhood made many discoveries.

To count the number of solutions to his equation would take a long time – an infinite time, in fact! There is also an infinite number of integer solutions, called Pythagorean triples.

These triples are related to each other by multiplication . The first triple in each related ‘family’ is called primitive.

Once you’ve mastered using the theorem, you might enjoy learning about other mathematicians! The story of Fermat’s last is particularly interesting and begins with a similar equation…

If you have a question or comment to leave below, don’t be shy! It’s helpful to learn from each other.

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Pythagoras - Pythagorean Problems Worksheet

Subject: Mathematics

Age range: 11-14

Resource type: Worksheet/Activity

Last updated

21 February 2021

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Students are to use Pythagoras’ Theorem to find the answers to real-life and problem solving questions using these differentiated worksheets.

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This is a bundle featuring the following worksheet resources: [Identifying the Hypotenuse](https://www.tes.com/teaching-resource/pythagoras-identifying-the-hypotenuse-12491115) [Calculating the Hypotenuse](https://www.tes.com/teaching-resource/pythagoras-calculating-the-hypotenuse-12491117) [Calculating a Shorter Side](https://www.tes.com/teaching-resource/pythagoras-calculating-a-shorter-side-12491118) [Calculating Missing Sides](https://www.tes.com/teaching-resource/pythagoras-calculating-missing-sides-12491120) [Is it a Right-Angled Triangle?](https://www.tes.com/teaching-resource/pythagoras-is-it-a-right-angled-triangle-12491122) [Distance Between Two Points](https://www.tes.com/teaching-resource/pythagoras-distance-between-two-points-worksheet-12491127) [Pythagorean Problems](https://www.tes.com/teaching-resource/pythagoras-pythagorean-problems-worksheet-12491130)

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