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Quizzes > High School Quizzes > Mathematics

Pythagorean Theorem Practice Quiz

Tackle engaging quiz questions and theorem word problems

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art promoting Pythagoras Theorem Showdown, a geometry trivia for high school students.

Use this Pythagorean Theorem quiz to practice solving right‑triangle word problems. Work through 20 quick questions with whole numbers, decimals, and real‑life situations, then see what feels easy and what needs review. Great for building speed and checking gaps before a class quiz or exam.

A right triangle has legs of lengths 3 cm and 4 cm. What is the length of the hypotenuse?
6 cm
8 cm
5 cm
7 cm
Using the Pythagorean theorem, 3² + 4² = 9 + 16 = 25. The square root of 25 is 5, which is the length of the hypotenuse.
Which of the following is the correct formula for the Pythagorean theorem in a right triangle?
a² + b² = c²
a² - b² = c²
a + b = c
a² + c² = b²
The Pythagorean theorem states that in a right triangle the sum of the squares of the two legs is equal to the square of the hypotenuse. The formula a² + b² = c² correctly expresses this relationship.
If one leg of a right triangle is 5 cm and the hypotenuse is 13 cm, what is the length of the other leg?
8 cm
11 cm
12 cm
10 cm
Applying the Pythagorean theorem: 5² + leg² = 13² gives 25 + leg² = 169. Subtracting 25 from 169 yields 144, so the other leg is the square root of 144, which is 12 cm.
What type of triangle is formed by sides measuring 6 cm, 8 cm, and 10 cm?
Obtuse Triangle
Isosceles Triangle
Equilateral Triangle
Right Triangle
Since 6² + 8² = 36 + 64 = 100, which equals 10², the triangle satisfies the Pythagorean theorem. Thus, it is a right triangle.
For a right triangle with legs measuring 9 cm and 12 cm, find the length of the hypotenuse.
16 cm
17 cm
18 cm
15 cm
By the Pythagorean theorem, 9² + 12² equals 81 + 144 = 225. Taking the square root of 225 gives 15, which is the length of the hypotenuse.
A 10-foot ladder leans against a wall. If the bottom of the ladder is 6 feet from the wall, how high does the ladder reach up the wall?
9 feet
7 feet
6 feet
8 feet
Using the Pythagorean theorem, the height reached is calculated as √(10² - 6²) = √(100 - 36) = √64, which equals 8 feet. This demonstrates finding an unknown side in a right triangle.
A right triangle has legs in the ratio 3:4 and a hypotenuse of 10 inches. What are the lengths of the legs?
5 inches and 9 inches
4 inches and 6 inches
3 inches and 7 inches
6 inches and 8 inches
Let the legs be 3k and 4k. The hypotenuse would then be 5k. With 5k = 10 inches, k equals 2, so the legs measure 6 inches and 8 inches. This is a classic Pythagorean triple.
In a right triangle, if one leg is 11 units and the hypotenuse is 15 units, what is the length of the other leg?
Approximately 9.3 units
Approximately 11.5 units
Approximately 8.5 units
Approximately 10.2 units
By applying the Pythagorean theorem, the missing leg is √(15² - 11²) = √(225 - 121) = √104, which is approximately 10.2 units. This demonstrates the use of square roots in solving right triangles.
In a right triangle, the sides are expressed as 5x, 12x, and 13x. Which of these represents the hypotenuse?
12x
13x
None of the above
5x
The hypotenuse is the longest side in a right triangle. Since 13x is greater than 5x and 12x, it is the hypotenuse according to the given ratio.
A rectangular field measures 30 meters by 40 meters. What is the length of the diagonal connecting opposite corners?
60 meters
50 meters
45 meters
40 meters
The diagonal forms the hypotenuse of a right triangle with legs measuring 30 m and 40 m. Using the Pythagorean theorem, the length is √(30² + 40²) = √(900 + 1600) = √2500 = 50 meters.
A right triangle has a hypotenuse of 17 cm and one leg measuring 8 cm. What is the length of the other leg?
12 cm
13 cm
16 cm
15 cm
Using the Pythagorean theorem, the other leg is calculated as √(17² - 8²) = √(289 - 64) = √225, which equals 15 cm. This completes the triangle's side lengths.
Find the hypotenuse of a right triangle with legs measuring 7 units and 24 units.
23 units
25 units
26 units
24 units
By applying the Pythagorean theorem, the hypotenuse is √(7² + 24²) = √(49 + 576) = √625, which equals 25 units. This is a classic example of a Pythagorean triple.
In a right triangle with sides of 9, 12, and x, what is the value of x?
21
10
15
3
Assuming 9 and 12 are the legs, the hypotenuse x is determined using the Pythagorean theorem: x = √(9² + 12²) = √(81 + 144) = √225, which is 15. This is a well-known Pythagorean triple.
If a right triangle has legs measuring 16 m and 30 m, what is the length of the hypotenuse?
34 m
36 m
32 m
46 m
Using the Pythagorean theorem, the hypotenuse is √(16² + 30²) = √(256 + 900) = √1156, which equals 34 m. This calculation confirms the triangle's dimensions.
A 20-foot ladder leaning against a wall reaches 16 feet high. What is the distance from the wall to the base of the ladder?
12 feet
10 feet
14 feet
8 feet
Using the Pythagorean theorem, the horizontal distance is determined as √(20² - 16²) = √(400 - 256) = √144, which equals 12 feet. This shows how to find an unknown side in a right triangle.
A square and a right triangle have the same perimeter; the triangle's sides are 9 cm, 12 cm, and 15 cm. What is the side length of the square?
9
10
8
7
The triangle's perimeter is 9 + 12 + 15 = 36 cm. Since the square has four equal sides, each side is 36/4, which equals 9 cm.
A ladder leans against a wall forming a right triangle. If a 15-foot ladder reaches 12 feet up the wall, how far is the base of the ladder from the wall?
10 feet
9 feet
7 feet
8 feet
By applying the Pythagorean theorem, the horizontal distance is √(15² - 12²) = √(225 - 144) = √81, which equals 9 feet. This is a practical application of the theorem in real life.
In a right triangle, if the hypotenuse measures 26 cm and one leg is twice the length of the other, what are the lengths of the legs?
Approximately 11.63 cm and 23.26 cm
Approximately 13 cm and 26 cm
Approximately 10 cm and 20 cm
Approximately 12 cm and 24 cm
Let the shorter leg be x and the longer leg be 2x. By the Pythagorean theorem, x² + (2x)² = 5x² equals 26² (676). Solving yields x ≈ 11.63 cm and 2x ≈ 23.26 cm.
Two right triangles share the same hypotenuse. One triangle has legs of 6 and 8, while the other has one leg measuring 5. What is the length of the missing leg in the second triangle?
Approximately 9.50
Approximately 10.00
Approximately 8.66
Approximately 7.50
The first triangle's hypotenuse is √(6² + 8²) = √(36 + 64) = √100, which is 10. For the second triangle, the missing leg is √(10² - 5²) = √(100 - 25) = √75, approximately 8.66.
In a right triangle with a perimeter of 60 cm, if one leg measures 20 cm and the hypotenuse is 25 cm, what is the length of the remaining leg?
30 cm
20 cm
25 cm
15 cm
Using the Pythagorean theorem, the missing leg is √(25² - 20²) = √(625 - 400) = √225, which equals 15 cm. Adding this to the other sides confirms the perimeter: 20 + 15 + 25 = 60 cm.
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Study Outcomes

  1. Apply the Pythagorean theorem to solve real-world geometry problems.
  2. Analyze right triangle properties to determine missing side lengths.
  3. Interpret problem statements and diagrams to identify relevant information.
  4. Evaluate multiple solution approaches for accuracy and efficiency.
  5. Demonstrate understanding of geometric relationships in practical scenarios.

Pythagorean Theorem Quiz & Word Problems Cheat Sheet

  1. Understand the Pythagorean Theorem - This classic rule tells you that in a right-angled triangle the square of the hypotenuse (the longest side) equals the sum of the squares of the other two sides. Think of it as a math magic trick where a² + b² always equals c²!
  2. Identify right-angled triangles in word problems - Spotting the right angle is your first step to victory in word problems. Keep an eye out for ladders leaning on walls, corners of buildings, or any time you see a perfect "L" shape.
  3. Apply the theorem to find missing side lengths - Given two sides, crank up the formula to find the third. For instance, if your ladder is 13 feet long and sits 5 feet from the wall, you can check if it reaches a 12-foot height like a pro.
  4. Use the theorem in real-life applications - From measuring the diagonal of a TV screen to figuring out the shortest walking path in a park, this theorem shows up everywhere. Turn everyday puzzles into quick calculations!
  5. Practice with diverse word problems - The more varied the scenarios, the sharper your skills become. Dive into physics-style distance questions or geometry puzzles to keep things interesting.
  6. Recognize Pythagorean triples - Memorize popular integer sets like (3, 4, 5) or (5, 12, 13) to instantly know you've got a right triangle. These number squads are your secret shortcut!
  7. Understand the converse of the Pythagorean Theorem - If a² + b² equals c², then voila, you have a right triangle. This reverse check is perfect for verifying angles in tricky geometry puzzles.
  8. Apply the theorem in coordinate geometry - Use it to calculate the distance between two points (x₝, y₝) and (x₂, y₂) via √[(x₂ - x₝)² + (y₂ - y₝)²]. It's like GPS for plotting points on graph paper!
  9. Solve problems involving three-dimensional figures - Extend to 3D by finding space diagonals of boxes: d = √(l² + w² + h²). Perfect for architects or anyone who dreams in 3D!
  10. Practice with worksheets and quizzes - Boost retention by tackling timed quizzes and printable worksheets. The more you practice, the more Pythagorean challenges feel like fun brain teasers!
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