# How to Master the Pythagorean Theorem and Right Triangles

Right triangles are central to geometry, chiefly because of their relationship with the Pythagorean Theorem. This theorem, which connects the lengths of the three sides of a right triangle, is fundamental for anyone delving into mathematical studies. In this guide, we will break down the concept of right triangles and elucidate the Pythagorean Theorem, providing a clear understanding of their mathematical significance and application.

## Step-by-step Guide: Right Triangles

**1. Definition of a Right Triangle:**

A right triangle is a triangle in which one of the angles measures exactly \(90^\circ\). This angle is termed the “right angle.” The side opposite the right angle is the longest side, known as the “hypotenuse”, while the other two sides are referred to as the “legs.”

**2. Pythagorean Theorem:**

For any right triangle, the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\) and \(b\)).

\( c^2 = a^2 + b^2 \)

**3. Special Right Triangles:**

Some right triangles have side ratios that consistently adhere to specific values:

**45-45-90 triangle**: This is an isosceles right triangle where the two legs are congruent, and the hypotenuse is \(\sqrt{2}\) times the length of one leg.**30-60-90 triangle**: In this triangle, the sides are in the ratio of \(1: \sqrt{3} : 2\), with the smallest side opposite the \(30^\circ\) angle and the longest side being the hypotenuse.

## Examples

**Example 1:**

Given a right triangle with legs measuring \(6 \text{ cm} \) and \(8 \text{ cm} \), find the length of the hypotenuse.

**Solution:**

Using the Pythagorean theorem:

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

\( c^2 = 36 + 64 = 100 \)

\( c = 10 \text{ cm} \)

**Example 2:**

Determine the length of the hypotenuse in a 45-45-90 triangle if one of the legs measures \(5 \text{ cm} \).

**Solution:**

For a 45-45-90 triangle, the hypotenuse is \(\sqrt{2}\) times the length of a leg:

\( c = 5 \times \sqrt{2} \approx 7.07 \text{ cm} \)

### Practice Questions:

- In a right triangle, if one leg measures \(9 \text{ cm} \) and the hypotenuse measures \(15 \text{ cm} \), find the length of the other leg.
- Calculate the longer leg in a 30-60-90 triangle if the shorter leg (opposite the \(30^\circ\) angle) measures \(4 \text{ cm} \).

**Answers:**

- Using the Pythagorean theorem, \( b^2 = 15^2 – 9^2 = 144 \) so \( b = 12 \text{ cm} \).
- The longer leg (opposite the \(60^\circ\) angle) is \(\sqrt{3}\) times the shorter leg, so it measures \(4 \times \sqrt{3} \approx 6.93 \text{ cm} \).

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