how to do pythagorean theorem word problems

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Course: 8th grade   >   Unit 5

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Pythagorean theorem word problems

• Pythagorean theorem in 3D

• Your answer should be
• an integer, like 6 ‍  
• an exact decimal, like 0.75 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  

Word problems on Pythagorean Theorem

Learn how to solve different types of word problems on Pythagorean Theorem .

Pythagoras Theorem can be used to solve the problems step-by-step when we know the length of two sides of a right angled triangle and we need to get the length of the third side.

Three cases of word problems on Pythagorean Theorem :

Case 1: To find the hypotenuse where perpendicular and base are given.

Case 2: To find the base where perpendicular and hypotenuse are given.

Case 3: To find the perpendicular where base and hypotenuse are given.

Word problems using the Pythagorean Theorem:

1. A person has to walk 100 m to go from position X in the north of east direction to the position B and then to the west of Y to reach finally at position Z. The position Z is situated at the north of X and at a distance of 60 m from X. Find the distance between X and Y.

⇒ 200x = 10000 + 3600

⇒ 200x = 13600

⇒ x = 13600/200

Therefore, distance between X and Y = 68 meters.

Therefore, length of each side is 8 cm.

Using the formula solve more word problems on Pythagorean Theorem.

3. Find the perimeter of a rectangle whose length is 150 m and the diagonal is 170 m.

In a rectangle, each angle measures 90°.

Therefore PSR is right angled at S

Using Pythagoras theorem, we get

⇒ PS = √6400

Therefore perimeter of the rectangle PQRS = 2 (length + width)

                                                          = 2 (150 + 80) m

                                                          = 2 (230) m

                                                          = 460 m

4. A ladder 13 m long is placed on the ground in such a way that it touches the top of a vertical wall 12 m high. Find the distance of the foot of the ladder from the bottom of the wall.

Let the required distance be x meters. Here, the ladder, the wall and the ground from a right-angled triangle. The ladder is the hypotenuse of that triangle.

According to Pythagorean Theorem,

Therefore, distance of the foot of the ladder from the bottom of the wall = 5 meters.

5. The height of two building is 34 m and 29 m respectively. If the distance between the two building is 12 m, find the distance between their tops.

The vertical buildings AB and CD are 34 m and 29 m respectively.

Draw DE ┴ AB

Then AE = AB – EB but EB = BC

Therefore AE = 34 m - 29 m = 5 m

Now, AED is right angled triangle and right angled at E.

⇒ AD = √169

Therefore the distance between their tops = 13 m.

The examples will help us to solve various types of word problems on Pythagorean Theorem.

Congruent Shapes

Congruent Line-segments

Congruent Angles

Congruent Triangles

Conditions for the Congruence of Triangles

Side Side Side Congruence

Side Angle Side Congruence

Angle Side Angle Congruence

Angle Angle Side Congruence

Right Angle Hypotenuse Side congruence

Pythagorean Theorem

Proof of Pythagorean Theorem

Converse of Pythagorean Theorem

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The pythagorean theorem with examples.

The Pythagorean theorem is a way of relating the leg lengths of a right triangle to the length of the hypotenuse, which is the side opposite the right angle. Even though it is written in these terms, it can be used to find any of the side as long as you know the lengths of the other two sides. In this lesson, we will look at several different types of examples of applying this theorem.

Table of Contents

• Examples of using the Pythagorean theorem
• Solving applied problems (word problems)
• Solving algebraic problems

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Applying the Pythagorean theorem (examples)

In the examples below, we will see how to apply this rule to find any side of a right triangle triangle. As in the formula below, we will let a and b be the lengths of the legs and c be the length of the hypotenuse. Remember though, that you could use any variables to represent these lengths.

In each example, pay close attention to the information given and what we are trying to find. This helps you determine the correct values to use in the different parts of the formula.

Find the value of \(x\).

The side opposite the right angle is the side labelled \(x\). This is the hypotenuse. When applying the Pythagorean theorem, this squared is equal to the sum of the other two sides squared. Mathematically, this means:

\(6^2 + 8^2 = x^2\)

Which is the same as:

\(100 = x^2\)

Therefore, we can write:

\(\begin{align}x &= \sqrt{100}\\ &= \bbox[border: 1px solid black; padding: 2px]{10}\end{align}\)

Maybe you remember that in an equation like this, \(x\) could also be –10, since –10 squared is also 100. But, the length of any side of a triangle can never be negative and therefore we only consider the positive square root.

In other situations, you will be trying to find the length of one of the legs of a right triangle. You can still use the Pythagorean theorem in these types of problems, but you will need to be careful about the order you use the values in the formula.

Find the value of \(y\).

The side opposite the right angle has a length of 12. Therefore, we will write:

\(8^2 + y^2 = 12^2\)

This is the same as:

\(64 + y^2 = 144\)

Subtracting 64 from both sides:

\(y^2 = 80\)

\(\begin{align}y &= \sqrt{80} \\ &= \sqrt{16 \times 5} \\ &= \bbox[border: 1px solid black; padding: 2px]{4\sqrt{5}}\end{align}\)

In this last example, we left the answer in exact form instead of finding a decimal approximation. This is common unless you are working on an applied problem.

Applications (word problems) with the Pythagorean theorem

There are many different kinds of real-life problems that can be solved using the Pythagorean theorem. The easiest way to see that you should be applying this theorem is by drawing a picture of whatever situation is described.

Two hikers leave a cabin at the same time, one heading due south and the other headed due west. After one hour, the hiker walking south has covered 2.8 miles and the hiker walking west has covered 3.1 miles. At that moment, what is the shortest distance between the two hikers?

First, sketch a picture of the information given. Label any unknown value with a variable name, like x.

Due south and due west form a right angle, and the shortest distance between any two points is a straight line. Therefore, we can apply the Pythagorean theorem and write:

\(3.1^2 + 2.8^2 = x^2\)

Here, you will need to use a calculator to simplify the left-hand side:

\(17.45 = x^2\)

Now use your calculator to take the square root. You will likely need to round your answer.

\(\begin{align}x &= \sqrt{17.45} \\ &\approx 4.18 \text{ miles}\end{align}\)

As you can see, it will be up to you to determine that a right angle is part of the situation given in the word problem. If it isn’t, then you can’t use the Pythagorean theorem.

Algebra style problems with the Pythagorean theorem

There is one last type of problem you might run into where you use the Pythagorean theorem to write some type of algebraic expression. This is something that you will not need to do in every course, but it does come up.

A right triangle has a hypotenuse of length \(2x\), a leg of length \(x\), and a leg of length y. Write an expression that shows the value of \(y\) in terms of \(x\).

Since no figure was given, your first step should be to draw one. The order of the legs isn’t important, but remember that the hypotenuse is opposite the right angle.

Now you can apply the Pythagorean theorem to write:

\(x^2 + y^2 = (2x)^2\)

Squaring the right-hand side:

\(x^2 + y^2 = 4x^2\)

When the problem says “the value of \(y\)”, it means you must solve for \(y\). Therefore, we will write:

\(y^2 = 4x^2 – x^2\)

Combining like terms:

\(y^2 = 3x^2\)

Now, use the square root to write:

\(y = \sqrt{3x^2}\)

Finally, this simplifies to give us the expression we are looking for:

\(y = \bbox[border: 1px solid black; padding: 2px]{x\sqrt{3x}}\)

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The Pythagorean theorem allows you to find the length of any of the three sides of a right triangle. It is one of those things that you should memorize, as it comes up in all areas of math, and therefore in many different math courses you will probably take. Remember to avoid the common mistake of mixing up where the legs go in the formula vs. the hypotenuse and to always draw a picture when one isn’t given.

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How to Solve Pythagorean Theorem Word Problems

The Pythagorean Theorem is a cornerstone of mathematics, fundamental in various scientific fields and real-life situations. We will guide you through understanding and solving Pythagorean Theorem word problems, explaining each step meticulously for your convenience.

The Pythagorean Theorem is an equation attributed to the ancient Greek mathematician, Pythagoras. The theorem states that in any right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The formula can be written as \(a^2+b^2=c^2\).

An essential part of understanding the Pythagorean Theorem involves recognizing Pythagorean triplets. These are sets of three integers that satisfy the Pythagorean Theorem. Notable examples are \((3,4,5), (5,12,13)\), and \((8,15,17)\). Recognizing these sets can simplify solving Pythagorean Theorem word problems.

A Step-by-step Guide to Solving Pythagorean Theorem Word Problems

Now that we’ve laid the groundwork, let’s delve into the process of solving word problems involving the Pythagorean Theorem.

Step 1: Analyzing the Problem

The first step in solving any word problem is understanding what the problem is asking. Translate the problem into an understandable format and identify the elements related to the Pythagorean Theorem.

Step 2: Sketching the Problem

Visualizing the problem can be extremely helpful. Sketch the triangle described in the problem and label the sides according to the given information. The right angle is usually denoted by a small square, and the hypotenuse, by the letter c .

Step 3: Applying the Pythagorean Theorem

Apply the Pythagorean Theorem to the problem. Plug in the values of the known sides into the theorem and solve for the unknown side.

Step 4: Verifying the Solution

After you’ve found the value of the unknown side, verify your solution by substituting the values back into the theorem. If the sides satisfy the theorem, you’ve correctly solved the problem.

Example of a Pythagorean Theorem Word Problem

Consider a problem where you’re given a right-angled triangle with one side measuring \(5\) units and the hypotenuse measuring \(13\) units. You’re asked to find the length of the other side.

You can use the Pythagorean Theorem to solve this problem. By substituting \(a=5\) and \(c=13\) into the theorem, you can solve for \(b\).

Calculating this will give you \(b^2=13^2-5^2=144\). Therefore, \(b=\sqrt{144}=12\) units. Your verification will involve substituting \(a=5, b=12\), and \(c=13\) into the theorem. The equation \(5^2+12^2=13^2\) checks out, confirming the solution.

The Pythagorean Theorem is a potent tool for solving geometrical problems involving right-angled triangles. With the steps outlined in this guide, you can confidently tackle any word problem that comes your way. Practice is key to mastery, so take time to solve different problems and apply the theorem in real-world situations.

by: Effortless Math Team about 12 months ago (category: Articles )

Effortless Math Team

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Pythagorean Theorem

How to Use The Pythagorean Theorem

The Formula

The picture below shows the formula for the Pythagorean theorem. For the purposes of the formula, side $$ \overline{c}$$ is always the hypotenuse . Remember that this formula only applies to right triangles .

Examples of the Pythagorean Theorem

When you use the Pythagorean theorem, just remember that the hypotenuse is always 'C' in the formula above. Look at the following examples to see pictures of the formula.

Conceptual Animation of Pythagorean Theorem

Demonstration #1.

More on the Pythagorean theorem

Demonstration #2

Video tutorial on how to use the pythagorean theorem.

Step By Step Examples of Using the Pythagorean Theorem

Example 1 (solving for the hypotenuse).

Use the Pythagorean theorem to determine the length of X.

Identify the legs and the hypotenuse of the right triangle .

The legs have length 6 and 8 . $$X $$ is the hypotenuse because it is opposite the right angle.

Substitute values into the formula (remember 'C' is the hypotenuse).

$ A^2+ B^2= \red C^2 \\ 6^2+ 8^2= \red X^2 $

$A^2+ B^2= \red X^2 \\ 100= \red X^2 \\ \sqrt {100} = \red X \\ 10= \red X $

Example 2 (solving for a Leg)

The legs have length 24 and $$X$$ are the legs. The hypotenuse is 26.

$ \red A^2+ B^2= C^2 \\ \red x^2 + 24^2= {26}^2 $

$ \red x^2 + 24^2= 26^2 \\ \red x^2 + 576= 676 \\ \red x^2 = 676 - 576 \\ \red x^2 = 100 \\ \red x = \sqrt { 100} \\ \red x = 10 $

Practice Problems

Find the length of X.

Remember our steps for how to use this theorem. This problems is like example 1 because we are solving for the hypotenuse .

The legs have length 14 and 48 . The hypotenuse is X.

$ A^2 + B^2 = C^2 \\ 14^2 + 48^2 = x^2 $

Solve for the unknown.

$ 14^2 + 48^2 = x^2 \\ 196 + 2304 = x^2 \\ \sqrt{2500} = x \\ \boxed{ 50 = x} $

Use the Pythagorean theorem to calculate the value of X. Round your answer to the nearest tenth.

Remember our steps for how to use this theorem. This problems is like example 2 because we are solving for one of the legs .

The legs have length 9 and X . The hypotenuse is 10.

$ A^2 + B^2 = C^2 \\ 9^2 + x^2 = 10^2 $

$ 9^2 + x^2 = 10^2 \\ 81 + x^2 = 100 \\ x^2 = 100 - 81 \\ x^2 = 19 \\ x = \sqrt{19} \approx 4.4 $

Use the Pythagorean theorem to calculate the value of X. Round your answer to the nearest hundredth.

The legs have length '10' and 'X'. The hypotenuse is 20.

$ A^2 + B^2 = C^2 \\ 10^2 + \red x^2 = 20^2 $

$ 10^2 + \red x^2 = 20^2 \\ 100 + \red x^2 = 400 \\ \red x^2 = 400 -100 \\ \red x^2 = 300 \\ \red x = \sqrt{300} \approx 17.32 $

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Top 7 real-life Pythagorean theorem word problems

• Author: Noreen Niazi
• Last Updated on: August 22, 2023

Do you know about the Pythagorean theorem?

Are you familiar with how to solve the Pythagorean theorem?

As a math student, the Pythagorean theorem is one of the first concepts you learn in high school. The theorem is a mathematical formula used to find the length of one side of a right triangle if the other two sides are known. However, solving Pythagorean theorem word problems can take time and effort. In this article, I will share the top seven Pythagorean theorem word problems you might encounter in real life and some tips for solving them.

Introduction to the Pythagorean Theorem

The Pythagorean theorem is a mathematical formula used to find the length of one side of a right triangle if the other two sides are known. The theorem states that the square of the hypotenuse (the longest side of the triangle) equals the sum of the squares of the other two sides. The Pythagorean theorem formula is written as a² + b² = c², where “a” and “b” are the two shorter sides of the triangle, and “c” is the hypotenuse[ 1] .

Steps to solve Pythagorean theorem word problems

Here are the steps you can take to solve Pythagorean theorem word problem s:

• Read the problem carefully and identify what is given and what needs to be found.
• Identify the sides of the triangle that are known and unknown.
• Write the formula for the Pythagorean theorem and substitute known values into the formula.
• Solve the equation to find the unknown value.
• Check your answer and make sure it makes sense with the problem.

Learn more about the percent word problems ,

Mastering Percent Word Problems In 2023: Tips And Tricks For Success – LearnAboutMath

Tips for Solving Pythagorean Theorem Word Problems

Here are some tips you can follow to solve Pythagorean theorem word problems more easily:

• Draw a diagram of the triangle and label the known and unknown sides.
• Be consistent with the measurement units used in the problem.
• Check that the triangle is a right triangle before using the Pythagorean theorem formula.
• If the value found for the length of one side of the triangle is not an integer, round to the nearest decimal number or answer in radical form.

Get more details about the Different math Problem solving strategies in Math

19 Best Math Problem-Solving Strategies For Elementary Students – LearnAboutMath

What are Pythagorean theorem word problems?

In word problems employing the Pythagorean theorem, the length of one side of a right triangle is determined using the formula. Finding the length of a side that is not the hypotenuse is common in these issues. Word problems are based on the Pythagorean Theorem in many facets of life, including science, engineering, and architecture.

Learn about the divison word probelms.

Mastering Long Division Word Problems: Tips And Tricks – LearnAboutMath

Understanding the Fundamentals of the Pythagorean Theorem Word Problems

To comprehend Pythagorean theorem word problems, you must first understand the theorem itself. According to the theorem, the hypotenuse’s square equals the sum of the squares of the triangle’s other two sides. In a triangle, the hypotenuse, opposite the right angle, is the longest side. The legs are the other two sides of the triangle.

Mastering Proportion Word Problems: A Step-by-Step Guide – LearnAboutMath

Real Life Pythagorean Theorem Word Problem - Problem 1

An architect is designing a house and needs to calculate the length of a hallway that connects two rooms. He knows that the first room has a wall that is 12 feet long, and the second room has a wall that is 9 feet long. The hallway is a right triangle. What is the length of the hallway?

To solve this problem, we need to use the Pythagorean theorem formula.

• We know that the first wall is 12 feet long, and the second is 9 feet long.
• Let’s call the unknown length of the corridor “x.”
• We can write the equation as

$$12² + 9² = x²$$

Solving the equation, we find that x² = 225, which means x = 15 feet.

Therefore, the length of the corridor is 15 feet.

Real Life Pythagorean Theorem Word Problem - Problem 2

An engineer is building a bridge and needs to calculate the cable length between two towers. He knows that the horizontal distance between the towers is 200 feet and the vertical distance is 75 feet. What is the cable length?

• We know the horizontal distance between the towers is 200 feet, and the vertical distance is 75 feet.
• Let’s call the unknown cable length “x.”
• We can write the equation as 200² + 75² = x².
• Solving the equation, we find that x² = 45.625, which means x = 213.8 feet (rounding to the nearest decimal number).

Therefore, the cable length is 213.8 feet.

Real Life Pythagorean Theorem Word Problem - Problem 3

A scientist is studying a volcano and needs to calculate the distance between his observation point and the volcano’s base.

• He knows that his position is 500 feet above sea level and that the volcano’s base is 1000 feet above sea level.
• He also knows that the horizontal distance between his position and the volcano’s base is 2000 feet.
• How far is your observation point from the base of the volcano?
• We know that the height of the observation point is 500 feet above sea level, and the volcano’s base is 1000 feet above sea level.
• Let’s call the unknown distance between the observation point and the volcano’s base “x.”
• The horizontal distance between the observation point and the volcano’s base is 2000 feet.
• We can write the equation as 500² + 2000² = (x + 1000)².
• Solving the equation, we find that x² = 3,500,000, which means that x = 1870 feet (rounding to the nearest decimal number).

Therefore, the distance between the observation point and the volcano’s base is 1870 feet

Real Life Pythagorean Theorem Word Problem - Problem 4

A farmer needs to build a fence around a rectangle to keep his animals safe.

• He knows that one side of the rectangle is 30 feet and that the diagonal of the rectangle is 50 feet.
• What is the length of the other side of the rectangle?
• We know that one side of the rectangle is 30 feet and that the diagonal of the rectangle is 50 feet.
• Let’s call the unknown length of the other side of the rectangle “x”.
• We can write the equation as 30² + x² = 50².
• Solving the equation, we find that x² = 1600, which means x = 40 feet.

Therefore, the length of the other side of the rectangle is 40 feet.

Real Life Pythagorean Theorem Word Problem - Problem 5

A pilot flying straight from one city to another must avoid a mountain.

• He knows that the mountain’s altitude is 1500 feet, and his flight altitude is 10,000 feet.
• He also knows the horizontal distance between the two cities is 50 miles.
• How far does he need to dodge to avoid the mountain?
• We know the mountain’s altitude is 1500 feet, and the pilot’s flight altitude is 10,000 feet.
• Let’s call the unknown distance the pilot needs to deviate “x.”
• We know that the horizontal distance between the two cities is 50 miles.
• We can convert this distance to feet by multiplying by 5280, which means the horizontal distance is 264,000 feet.
• We can write the equation as 1500² + x² = 10,000².
• Solving the equation, we find that x² = 98,551,000, which means that x = 9,927 feet (rounding to the nearest decimal number).

Therefore, the pilot needs to deviate 9,927 feet to avoid the mountain.

Real Life Pythagorean Theorem Word Problem - Problem 6

A diver is diving in a lake and must swim to a buoy 30 feet away.

• He knows that the depth of the lake is 20 feet.
• How far does he have to swim to reach the buoy?
• We know the lake is 20 feet deep, and the buoy is 30 feet away.
• The unknown distance the diver needs to swim “x”.
• We can write the equation as 20² + 30² = x².
• Solving the equation, we find that x² = 1300, which means x = 36.06 feet (rounding to the nearest decimal number).

Therefore, the diver needs to swim 36.06 feet to reach the buoy.

Real Life Pythagorean Theorem Word Problem - Problem 7

An archaeologist is measuring the height of an ancient pyramid.

• He knows that the base of the pyramid is 100 feet long and that the pyramid is 75 feet high.
• What is the length of a straight line from the top of the pyramid to the ground?
• We know that the base of the pyramid is 100 feet long and that the pyramid is 75 feet high.
• Let’s call the unknown length of the straight line from the top of the pyramid to the ground “x”.
• We can write the equation as 75² + (50)² = x².
• Solving the equation, we find that x² = 7.625, which means x = 87.3 feet (rounding to the nearest decimal number).

Therefore, the straight line measurement from the top of the pyramid to the ground is 87.3 feet.

'Conclusion and Tips to Solve Pythagorean Theorem Word Problems

The Pythagorean theorem makes finding a right triangle’s missing side or the angle between two sides easier. According to this rule, the square of the hypotenuse, the longest side, equals the sum of the squares of the other two sides. The right angle must be located, the sides must be denoted by the letters a, b, and c, and the known values must be entered into the equation $a2 + b2 = c2$ to apply the theorem. We can then use trigonometric functions to get the angle or solve for the unknown side.

The following are some pointers for resolving word problems with the Pythagorean theorem:

– Sketch a circumstance diagram, labeling the sides and angles as necessary. – Determine if the triangle is right-angled by looking for hints in the problem or applying the Pythagorean theorem’s opposite. – To get the omitted side or angle, apply the Pythagorean theorem or one of its variations, such as $$a2 + b2 – 2ab cos C = c2$$. – Your response should be rounded to the proper degree of precision and, if necessary, include units. – Put your solution back into the formula or use another approach to verify it.

FAQs: Pythagorean Theorem Word Problems

The following actions must be taken in order to use the Pythagorean theorem to solve word problems:

• Name the sides of the right triangle in the given situation as a, b, and c.
• Choose the sides that you have been provided and that you need to find.
• Solve for the unknown side by substituting the provided values into the Pythagorean theorem.
• Round off your response to the required degree of precision.

The Pythagorean theorem can be applied in a variety of circumstances to determine the shortest distance, sound wave speed, square angles, and the missing side lengths of right triangles.

Here is an real life example of Pythagorean theorem.

Navigation: The Pythagorean theorem can be used to determine the quickest route and the direction to take if you are sailing or flying and wish to reach a destination that is not directly north, south, east, or west of your current location.

The Pythagorean theorem, for instance, can be used to determine that the shortest distance and direction are approximately 500 miles and 53 degrees west of north, respectively, if you are at sea and travelling to a position that is 300 miles north and 400 miles west of your present location.

Finding the length of a right triangle’s third side given the measurements of its other two sides is an illustration of the Pythagorean theorem in mathematics. A triangle with a single angle that measures 90 degrees is referred to as a right triangle. The hypotenuse is the side of a right triangle that is the longest and sits across from the right angle.

The Pythagorean theorem can be used to determine the length of the hypotenuse, for instance, if a right triangle has legs that are 3 cm and 4 cm long:

Replace the supplied values in the equation as follows: Simplify: 3 + 4 = c 2. 9 + 16 = c 2.2. Add: 25 = c^2. Determine the square root of each side: c = 25 = 5. The hypotenuse is 5 cm long as a result.

[1].  https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-geometry/pythagorean-theorem-application/e/pythagorean-theorem-word-problems–basic

[2]. https://study.com/skill/learn/how-to-solve-pythagorean-theorem-word-problems-explanation.html -fractions/#:~:text=An%20improper%20fraction%20has%20a,is%20greater%20than%20the%20numerator .

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15 Pythagorean Theorem Practice Problems For 8th Grade

Beki Christian

Pythagorean Theorem practice problems involve using the relationship between the sides of a right triangle to calculate missing side lengths in triangles. The Pythagorean Theorem is introduced in 8th grade and is used to solve a variety of problems across high school.

Here, you’ll find a selection of Pythagorean Theorem questions that demonstrate the different types of questions students are likely to encounter in 8th grade.

What is the Pythagorean Theorem?

The Pythagorean Theorem is the geometric theorem that states that the square of the hypotenuse (longest side) of a right triangle is equal to the sum of the squares of the two shorter sides of the triangle.

This can be written as a^2+b^2=c^2 for a triangle labeled like this:

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How to answer Pythagorean Theorem questions

1 – Label the sides of the triangle a , b , and c . Note that the hypotenuse, the longest side of a right triangle, is opposite the right angle and will always be labeled .

2 – Write down the formula and substitute the values>

3 – Calculate the answer. You may be asked to give your answer in an exact form or round to a given degree of accuracy, such as a certain number of decimal places or significant figures.

Pythagorean Theorem in real life

Pythagorean Theorem has many real-life uses, including in architecture and construction, navigation and surveying.

Pythagorean Theorem in 8th grade

Pythagorean Theorem is usually introduced in middle school, as it is a part of the 8th grade Common Core Math Standards.

The emphasis in middle school is on students being able to:

• Explain the Pythagorean Theorem;
• Use the theorem to solve mathematical and real-world problems – with both 2D and 3D figures;
• Use the theorem to calculate the distance between two points on a coordinate grid.

The process for solving any Pythagoras Theorem problem always begins by identifying the relevant right-angled triangle and labeling the sides a , b , c. If there is not a diagram in the question, it can be helpful to draw one.

Where necessary, round your answers to 3 significant figures.

Pythagorean Theorem practice problems

1. A ship sails 6 \, km East and then 8 \, km North. Find the ship’s distance from its starting point.

The ship is 10 kilometers from its starting point.

2. A ladder is 5 \, m long. The base of the ladder is 3 \, m from the base of a vertical wall. How far up the wall does the ladder reach?

The ladder reaches 4 meters up the wall.

3. Alex and Sam start from the same point. Alex walks 400 meters west. Sam walks x meters south, until they are 600 \, m apart from each other. How far does Sam walk?

4. A television’s size is the measurement from the upper left hand corner of the television to the bottom right hand corner. Find the size of this television.

39.7 inches

55.1 inches

5. The pole of a sailing boat is supported by a rope from the top of the pole to an anchor point on the deck. The pole is 4 \, m long and the rope is 4.5 \, m long. Calculate the distance from the base of the pole to the anchor point of the rope on the deck.

6. Work out the length of the diagonal of a square with 8 \, cm sides.

The diagonal of the square has a length of 11.3 centimeters.

7. ABC is an isosceles triangle.

Work out the height of the triangle.

8. ABCD is an isosceles trapezoid.

Work out the length of AD.

9. Here is a cm square grid. Calculate the distance between the points A and B.

10. Which is a right angled triangle?

Not a right angled triangle because Pythagorean Theorem doesn’t work.

Right angled triangle because Pythagorean Theorem works.

11. PQRS is made from two right angled triangles.

Work out the length of QR.

Triangle \text{PQS:}

Triangle \text{QRS}

12. Here is a pattern made from right angled triangles. Work out the length x.

Triangle \text{ABC:}

Triangle \text{ACD:}

13. Here is a pyramid.

Work out the height of the pyramid.

14. Here is a cuboid.

Work out the length AG.

Give your answer in its exact form.

Length of \text{BG:}

Length of \text{AG:}

15. Here is a right angled triangle.

Form an equation and use it to work out the value of x.

x=4 \, or \, x=12

x cannot be 4 as you can’t have a negative side length so x=12

Pythagorean Theorem in middle school

In middle school, students…

• prove the Pythagorean Theorem;
• use the Pythagorean Theorem with trigonometric ratios to solve problems;
• use the Pythagorean Theorem in proofs.

Pythagoras Theorem may feature in questions alongside other topics, such as trigonometry, circle theorems or algebra.

The Pythagorean Theorem is used to calculate a missing length in a right triangle . If you have a right angled triangle and you know two of the lengths, label the sides of the triangle a,b and c (c must be the hypotenuse – the longest side). Pythagorean Theorem is a^2+b^2=c^2. Substitute the values you know into Pythagorean Theorem and solve to find the missing side.

The hypotenuse of a right triangle is the longest side. If you know the lengths of the other two sides, you can find the length of the hypotenuse by squaring the two shorter sides, adding those values together and then taking the square root. By doing this you are finding c in a^2+b^2=c^2

If your triangle is a right triangle and you know two of the sides, you can use Pythagorean Theorem to find the length of the third side. To do this, label the sides a , b and c (with c being the hypotenuse – the longest side). Substitute the values you know into a^2+b^2=c^2 and solve to find the missing side.

Looking for more Pythagorean theorem math questions?

• Ratio questions
• Algebra questions
• Probability questions
• Trigonometry questions
• Venn diagram questions
• Long division questions

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade tutoring – 3rd grade tutoring – 4th grade tutoring – 5th grade tutoring – 6th grade tutoring – 7th grade tutoring – 8th grade tutoring Why not learn more about how it works ?

The content in this article was originally written by former UK Secondary teacher Beki Christian and has since been revised and adapted for US schools by elementary and middle school teacher Kathleen Epperson.

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Math Mammoth Square Roots & the Pythagorean Theorem Paperback – May 27, 2024

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Math Mammoth Square Roots & The Pythagorean Theorem is a relatively short worktext focusing on irrational numbers, square roots, and the Pythagorean Theorem and its applications.

First, students learn about taking a square root as the opposite operation to squaring a number. They learn about irrational numbers, and how to find approximations to square roots both with a calculator and with a guess-and-check method. Students also practice placing irrational numbers on the number line, using mental math to find their approximate location.

Next, the book has a review lesson on how to convert fractions to decimals. The following lesson has to do with writing decimals as fractions, and teaches a method for converting repeating decimals to fractions.

Then it is time to learn to solve simple equations that involve taking a square or cube root, over the course of two lessons. After learning to solve such equations, students are now fully ready to study the Pythagorean Theorem and apply it.

The Pythagorean Theorem is introduced in the lesson by that name. Students learn to verify that a triangle is a right triangle by checking whether it fulfills the Pythagorean Theorem. They apply their knowledge about square roots and solving equations to solve for an unknown side in a right triangle when two of the sides are given.

Next, students solve a variety of geometric and real-life problems that require the Pythagorean Theorem. This theorem is extremely important in many practical situations. Students should show their work for these word problems to include the equation that results from applying the Pythagorean Theorem to the problem and its solution.

There are literally hundreds of proofs for the Pythagorean Theorem. In this book, we present one easy proof based on geometry (not algebra). As an exercise, students are asked to supply the steps of reasoning to another geometric proof of the theorem. Students also study a proof for the converse of the theorem, which says that if the sides of a triangle fulfill the equation a 2 + b 2 = c 2 then the triangle is a right triangle.

Our last topic is distance between points in the coordinate grid, as this is another simple application of the Pythagorean Theorem.

• Print length 78 pages
• Language English
• Publisher Math Mammoth
• Publication date May 27, 2024
• Dimensions 8.5 x 0.16 x 11 inches
• ISBN-10 1954358741
• ISBN-13 978-1954358744
• See all details

Product details

• Publisher ‏ : ‎ Math Mammoth (May 27, 2024)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 78 pages
• ISBN-10 ‏ : ‎ 1954358741
• ISBN-13 ‏ : ‎ 978-1954358744
• Item Weight ‏ : ‎ 7.2 ounces
• Dimensions ‏ : ‎ 8.5 x 0.16 x 11 inches

About the author

Maria miller.

Maria Miller was born and grew up in Finland where she also received a master's degree in mathematics in 1997. She loves math and teaching, so becoming a math teacher was a natural career for her.

Miller started writing math books in 2002, prompted by her experiences in tutoring American homeschooled children. She felt the books they were using were inferior, especially when it came to conceptual understanding of mathematics, so she began to write books that focus on the concepts and the "why" of mathematics, though not forgetting the "how" and the needful practice.

Over the years, her materials have become popular and appreciate, not only among homeschooling parents but also teachers and parents who do not homeschool.

In her free time, she likes walking, bicycling, swimming, reading, helping out with Inspire4.com, playing piano, and listening to music.

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