Pythagorean Theorem Practice Problems and Worksheets

Master the Pythagorean theorem with step-by-step practice problems. Find missing sides in right triangles using a² + b² = c² formula with detailed solutions.

📚What You'll Master in These Pythagorean Theorem Practice Problems
  • Apply the formula c² = a² + b² to find missing hypotenuse lengths
  • Calculate unknown leg lengths when given hypotenuse and one leg
  • Solve isosceles right triangle problems with equal leg measurements
  • Use the converse theorem to verify if triangles are right triangles
  • Work through real-world applications like ramps and distance problems
  • Master square root calculations and algebraic manipulation techniques

Understanding Pythagorean Theorem

Complete explanation with examples

The Pythagorean Theorem can be formulated as follows: in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

In the right triangle shown in the image below, we use the first letters of the alphabet to indicate its sides:

a a and b b are the legs.

cc is the hypotenuse.

Using these, we can express the Pythagorean theorem in an algebraic form as follows:

c2=a2+b2 c²=a²+b²

How to solve the Pythagorean theorem

We can express the Pythagorean Theorem in a geometric form in the following way, showing that the area of the square (c c ) (square of the hypotenuse) is the sum of the areas of the squares (a a ) and (b b ) (squares of the legs).

geometrical form of the Pythagorean Theorem

Detailed explanation

Practice Pythagorean Theorem

Test your knowledge with 40 quizzes

Given the triangles in the drawing

What is the length of the side DB?

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Examples with solutions for Pythagorean Theorem

Step-by-step solutions included
Exercise #1

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What is the length of the hypotenuse?

Step-by-Step Solution

We use the Pythagorean theorem

AC2+AB2=BC2 AC^2+AB^2=BC^2

We insert the known data:

32+42=BC2 3^2+4^2=BC^2

9+16=BC2 9+16=BC^2

25=BC2 25=BC^2

We extract the root:

25=BC \sqrt{25}=BC

5=BC 5=BC

Answer:

5

Video Solution
Exercise #2

Look at the triangle in the diagram. How long is side AB?

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Step-by-Step Solution

To find side AB, we will need to use the Pythagorean theorem.

The Pythagorean theorem allows us to find the third side of a right triangle, if we have the other two sides.

You can read all about the theorem here.

Pythagorean theorem:

A2+B2=C2 A^2+B^2=C^2

That is, one side squared plus the second side squared equals the third side squared.

We replace the existing data:

32+22=AB2 3^2+2^2=AB^2

9+4=AB2 9+4=AB^2

13=AB2 13=AB^2

We find the root:

13=AB \sqrt{13}=AB

Answer:

13 \sqrt{13} cm

Video Solution
Exercise #3

Look at the triangle in the diagram. Calculate the length of side AC.

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Step-by-Step Solution

To solve the exercise, we have to use the Pythagorean theorem:

A²+B²=C²

 

We replace the data we have:

3²+4²=C²

9+16=C²

25=C²

5=C

Answer:

5 cm

Video Solution
Exercise #4

Consider a right-angled triangle, AB = 8 cm and AC = 6 cm.
Calculate the length of side BC.

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Step-by-Step Solution

To find the length of the hypotenuse BC in a right-angled triangle where AB and AC are the other two sides, we use the Pythagorean theorem: c2=a2+b2 c^2 = a^2 + b^2 .

Here, a=6 cm a = 6 \text{ cm} and b=8 cm b = 8 \text{ cm} .

Plugging the values into the Pythagorean theorem, we get:

c2=62+82 c^2 = 6^2 + 8^2 .

Calculating further:

c2=36+64 c^2 = 36 + 64

c2=100 c^2 = 100 .

Taking the square root of both sides gives:

c=10 cm c = 10 \text{ cm} .

Answer:

10 cm

Exercise #5

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What is the length of the side marked X?

Step-by-Step Solution

We use the Pythagorean theorem:

AB2+BC2=AC2 AB^2+BC^2=AC^2

Answer:

15 15

Video Solution

Frequently Asked Questions

How do I know which side is the hypotenuse in a right triangle?

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The hypotenuse is always the longest side of a right triangle and is located opposite the right angle (90° angle). It's the side you're looking for when you see the small square symbol indicating the right angle in triangle diagrams.

What's the difference between finding the hypotenuse and finding a leg?

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When finding the hypotenuse, you add the squares of both legs: c² = a² + b². When finding a leg, you subtract: a² = c² - b² or b² = c² - a². Always identify what you're solving for first.

Can I use the Pythagorean theorem on any triangle?

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No, the Pythagorean theorem only works on right triangles (triangles with a 90° angle). If you try to use it on acute or obtuse triangles, you won't get correct results.

What are the most common Pythagorean theorem mistakes students make?

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Common mistakes include: 1) Forgetting to take the square root at the end, 2) Using the theorem on non-right triangles, 3) Mixing up which side is the hypotenuse, 4) Making calculation errors with squares and square roots.

How do I solve Pythagorean theorem word problems step by step?

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Follow these steps: 1) Draw and label a right triangle from the problem, 2) Identify the legs and hypotenuse, 3) Substitute known values into a² + b² = c², 4) Solve the equation algebraically, 5) Take the square root if needed, 6) Check your answer makes sense.

What should I do when I get a decimal answer in Pythagorean theorem problems?

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Decimal answers are normal, especially when dealing with square roots like √50 = 7.07. You can leave answers in exact form (like √50) or round to a specified number of decimal places as requested in the problem.

How is the Pythagorean theorem used in real life?

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Real-world applications include: calculating distances on maps, determining ladder placement angles, designing ramps and stairs, construction and engineering projects, navigation and GPS systems, and sports field measurements.

What's the converse of the Pythagorean theorem and when do I use it?

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The converse states: if a² + b² = c² in a triangle, then it's a right triangle. Use this to verify whether a triangle is a right triangle when you know all three side lengths but aren't told if there's a right angle.

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