# Pythagorean Theorem Lesson

The Pythagorean Theorem is a fundamental principle in geometry that describes the relationship between sides of a right triangle and the hypotenuse. It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the lengths of the legs of the triangle.

In the formula below for the **Pythagorean Theorem**, a and b represent the lengths of the legs, and c represents the hypotenuse.

This formula can also be rearranged to solve for the length of the hypotenuse.

The formula for the Pythagorean Theorem can be applied to solve for a missing side length of any right triangle. If you know the lengths of both legs, you can find the length of the hypotenuse. Similarly, if you know the length of one leg and the hypotenuse, you can find the length of the other leg.

The theorem is named after the ancient Greek mathematician Pythagoras, who is credited with the theorem’s introduction. The Pythagorean Theorem is one of the most important and widely used theorems in mathematics, with applications in several fields. It is used to solve problems related to distance, determining whether a triangle is a right triangle, and finding unknown side lengths in right triangles.

All right triangles follow the rule of the Pythagorean Theorem, and the three side lengths that form any given right triangle are referred to as a Pythagorean Triple. There are some common Pythagorean Triples, such as a 3, 4, 5 right triangle or 12, 13, 15.

**Examples**

**Example 1**: Given a right triangle with a side length of 6 and the hypotenuse 10. Find the measure of the other side length.

**Solution**: Since we are told that it is a right triangle, we know that we can apply the Pythagorean Theorem. We recognize that we are given a leg length and the hypotenuse, therefore we know that:

a^{2} + (6)^{2} = (10)^{2}

a^{2} = 36 = 100

a^{2} = 100 – 36

a^{2} = 64

a = √64 = 8

Hence, the length of the other side of the right triangle is 8.

**Example 2: **Given a right triangle with side lengths of a = 8 and b = 15, find the length of the hypotenuse.

**Solution:** Since we are told it is a right triangle, we will apply the Pythagorean Theorem. Since our hypotenuse is our unknown, we will be solving for c in the equation.

8^{2} + 15 = c^{2}

64 + 225 = c^{2}

289 = c^{2}

c = √289 = 17

Therefore, the length of the hypotenuse is 17.

**Example 3: **Given the right triangle below, find the missing length.

**Solution:** Since we are told that it is a right triangle, we know that we can apply the Pythagorean Theorem. We recognize that we are given a leg length and the hypotenuse, therefore we will be solving for the missing leg length.

7^{2} + b^{2} = 25^{2}

49 + b^{2} = 625

b^{2} = 576

b = √576 = 24

FAQs on Pythagorean Theorem

**1) What is the Pythagorean Theorem?**

The Pythagorean Theorem states that in a right-angled triangle, 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, it can be expressed as: a^{2} + b^{2} = c^{2}, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

**2) Who discovered the Pythagorean Theorem?**

The Pythagorean Theorem is attributed to the ancient Greek mathematician Pythagoras, who lived in the 6th century BCE. However, it is possible that the theorem was known to other civilizations before Pythagoras.

**3) What are the applications of the Pythagorean Theorem?**

The Pythagorean Theorem has numerous applications in various fields, including:

**Geometry: **It is used to find the lengths of sides in right-angled triangles.

**Trigonometry:** It forms the basis for trigonometric functions such as sine, cosine, and tangent.

**Physics: **It is used in mechanics, optics, and other areas to calculate distances, velocities, and forces.

**Engineering:** It is used in construction, architecture, and design to ensure stability and accuracy in measurements.

**4) Can the Pythagorean Theorem be used in non-right-angled triangles**

No, the Pythagorean Theorem only applies to right-angled triangles. In non-right-angled triangles, other trigonometric principles such as the Law of Sines and the Law of Cosines are used to find the relationships between sides and angles.

**5) How can I prove the Pythagorean Theorem?**

There are various proofs of the Pythagorean Theorem, including geometric proofs, algebraic proofs, and even proofs using calculus. One of the most famous geometric proofs involves dissecting a square into four identical right triangles and a smaller square. Algebraic proofs often involve manipulating the algebraic expressions representing the areas of the squares formed on each side of the triangle.

**6) Are there any real-life examples of the Pythagorean Theorem**

Yes, the Pythagorean Theorem is applied in many real-life situations. For example: In construction, it is used to ensure that structures are built with accurate right angles and dimensions.

In navigation, it is used to calculate distances between points on a map or between celestial bodies.

In sports such as baseball or football, it can be used to measure distances on the field or the trajectory of a ball.

**7) What are some Pythagorean triples?**

Pythagorean triples are sets of three positive integers a, b, and c that satisfy the Pythagorean Theorem (a^{2} + b^{2} = c^{2}). Some common Pythagorean triples include (3, 4, 5), (5, 12, 13), and (8, 15, 17). These triples represent the side lengths of right-angled triangles with integer values.