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Pythagorean Theorem Trivia Quiz

The Pythagorean theorem (or Pythagoras' theorem) describes the relationship between the lengths of three sides of any right-angled triangle. This quiz tests your knowledge on the history, proof and application of this famous theorem. Enjoy!

A multiple-choice quiz by Matthew_07. Estimated time: 5 mins.

Author
Matthew_07

Time
5 mins

Type
Multiple Choice

Quiz #
325,214

Updated
Jan 20 22

# Qns
10

Difficulty
Average

Avg Score
7 / 10

Plays
987

Awards
Top 35% Quiz

One at a Time - Single Page - Timed Game

Question 1 of 10

1. The Pythagorean theorem was named after Pythagoras, who was a Greek mathematician and philosopher. The theorem states that for a right-angled triangle, the sum of the squares of the 2 shorter sides is equal to the square of length of the longest side. Let a and b denote the 2 shorter sides and c denote the longest side. Hence, this theorem can be represented by which of the following mathematical equations? Hint

a^2 + b^2 = c^2

a^2 = b^2 + c^2

(a + b)^2 = c^2

(a + c)^2 = b^2

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Question 2 of 10

2. The longest side of a right-angled triangle is the side opposite the right angle. By using the Pythagorean theorem, the length of the longest side can be calculated by using the formula h = square root of (f^2 + g^2), where f and g are the lengths of the two shorter sides. What is the specific mathematical term that is used to denote this longest side? Hint

Hypertangent

Hypotenuse

Hyperbola

Hypothesis

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Question 3 of 10

3. In trigonometry, the Pythagorean theorem is considered a special case of a more generic law. In other words, this mathematical law is a more general formula that can be used for all types of triangles. What is the name of this law? Hint

Law of cosines

Law of tangents

Law of cosecants

Law of sines

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Question 4 of 10

4. True or false: There is only one way to prove the Pythagoras theorem.

True

False

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Question 5 of 10

5. The 20th President of the United States provided an algebraic proof of the Pythagorean theorem. Who is he? Hint

Abraham Lincoln

George Washington

James A. Garfield

John F. Kennedy

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Question 6 of 10

6. Given a right-angled triangle and the lengths of its three sides are 3, 4 and 5. These three positive integers are known as a Pythagorean triple denoted by (3, 4, 5), as they satisfy the equation stated by the Pythagorean theorem. If I multiply each of the number by 10, will the resulting numbers, namely 30, 40 and 50 be another Pythagorean triple?

Yes

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Question 7 of 10

7. There are many formulae that can be used to generate Pythagorean triples. One of them states that a = m^2 - n^2, b = 2mn, c = m^2 + n^2 form a Pythagorean triple given that three conditions are met. First, the greatest common divisor of m and n is 1; second, only m or only n can be even; and third, m must be greater than n. What is the name of this formula? Hint

Plato's formula

Monet's formula

Archimedes' formula

Euclid's formula

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Question 8 of 10

8. A Pythagorean prime is any prime number that is also the length of the hypotenuse of a Pythagorean triple . The first few Pythagorean primes are 5, 13, 17, 29... What is the formula used to generate Pythagorean primes? (given that n is a subset of positive integers) Hint

3n + 2

4n + 1

2n + 3

n + 4

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Question 9 of 10

9. Which of the following coordinate systems uses the Pythagorean theorem to find the distance between 2 points? Hint

Elliptic coordinate system

Polar coordinate system

Cartesian coordinate system

Parabolic coordinate system

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Question 10 of 10

10. The Pythagorean theorem can only be applied to which of the following triangles?
Triangle A: a triangle that is inscribed in a semicircle, where the diameter of the semicircle is its longest side
Triangle B: an equilateral triangle
Triangle C: a scalene triangle, with length 7, 8 and 9
Triangle D: a triangle with length 1, 1 and square root of 2 Hint

Triangle A and D

Triangle A and C

Triangle B and C

Triangle B and D

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Quiz Answer Key and Fun Facts

Answer: a^2 + b^2 = c^2

Given the lengths of the 2 shorter sides of a triangle as a and b, it should be noted that the sum of the squares of the 2 sides is a^2 + b^2 while the square of the sum of the 2 sides is (a + b)^2. Algebraically, the correct mathematical equation is a^2+ b^2 = c^2.

Answer: Hypotenuse

The word "hypotenuse" is of Greek origin (hypoteinousa). The prefix "hypo-" means under while "tenuse" comes from the word "tenein", which means to stretch.

Answer: Law of cosines

The law of cosines is given by the following equation: c^2 = a^2 + b^2 - 2abcos gamma, where a, b, c are the lengths of the 3 sides of a triangle and gamma is the angle opposite side c.

Notice that when gamma = 90 degrees, cos 90 = 0. The equation simplifies to c^2 = a^2 + b^2, which is the formula of the Pythagorean theorem.

4. True or false: There is only one way to prove the Pythagoras theorem.

Answer: False

The Pythagorean theorem is unique in such a way that it can be proved by many different ways.

The book "The Pythagorean Proposition", written by Elisha S. Loomis contains as many as 367 proofs.

This page contains 84 different proofs: http://www.cut-the-knot.org/pythagoras/

5. The 20th President of the United States provided an algebraic proof of the Pythagorean theorem. Who is he?

Answer: James A. Garfield

In his proof, Garfield divided a trapezoid into 3 sections, namely 2 identical right-angled triangles of equal area and another isosceles triangle. By performing some algebraic manipulation, he arrived at the equation a^2 + b^2 = c^2.

Answer: Yes

The Pythagorean theorem states that a^2 + b^2 = c^2.

Note that 3^2 + 4^2 = 9 + 16 = 25 = 5^2. Also, 30^2 + 40^2 = 900 + 1600 = 2500 = 50^2.

In general, if (a, b, c) is a Pythagorean triple, then (pa, pb, pc) is also a Pythagorean triple, where p is a positive integer.

Answer: Euclid's formula

Euclid was a Greek mathematician. He was also known as the "Father of Geometry".

Let's take m = 3 and n = 2. Now, we have a = 3^2 - 2^2 = 9 - 4 = 5, b = 2 x 3 x 2 = 12 and c = 3^2 + 2^2 = 9 + 4 = 13. Therefore, the Pythagorean triple generated is (5, 12, 13) and we can verify this by using the equation a^2 + b^2 = c^2. Note that 5^2 + 12^2 = 25 + 144 = 169 = 13^2.

Answer: 4n + 1

The first few Pythagorean triples are (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25). Note that 4 x 1 + 1 = 5, 4 x 3 + 1 = 13, 4 x 4 + 1 = 17 and 4 x 6 + 1 = 25. Here, n = 1, 3, 4 and 6. Hence, the general formula that is used to generate Pythagorean primes is given by 4n + 1. Some values of n, such as 2 and 5 will NOT yield Pythagorean primes.

9. Which of the following coordinate systems uses the Pythagorean theorem to find the distance between 2 points?

Answer: Cartesian coordinate system

Given any 2 points (x1, y1) and (x2, y2) on a 2-D Cartesian plane, the formula that is used to calculate the distance between the 2 points is given by d = square root of [(x2 - x1)^2 + (y2 - y1)^2].

d^2 = (x2 - x1)^2 + (y2 - y1)^2

By replacing e^2 = (x2 - x1)^2 and f^2 = (y2 - y1)^2, we get d^2 = e^2 + f^2, which is exactly the same as the formula stated by the Pythagorean theorem.

10. The Pythagorean theorem can only be applied to which of the following triangles? Triangle A: a triangle that is inscribed in a semicircle, where the diameter of the semicircle is its longest side Triangle B: an equilateral triangle Triangle C: a scalene triangle, with length 7, 8 and 9 Triangle D: a triangle with length 1, 1 and square root of 2

Answer: Triangle A and D

We are looking for right-angled triangles here.

Triangle A is a right-angled triangle.

The interior angle of triangle B is 60 degrees.

A scalene triangle is a triangle where its 3 sides are of different lengths. It may or may not be a right-angled triangle. For triangle C, the three numbers do not satisfy the equation a^2 + b^2 = c^2, so it is not a right-angled triangle.

For triangle D, it is a right-angled triangle because 1^2 + 1^2 = 1 + 1 = 2 = [square root of 2]^2.

Source: Author Matthew_07

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