Grade 8

Intermediate

7 min

Lesson 8 of 12

The Pythagorean Theorem

Not Started

Introduction to the relationship between the sides of a right triangle.

What if you could calculate the exact distance of a shortcut across a park without ever stepping off the sidewalk? This 2,500-year-old secret allows architects, gamers, and GPS systems to measure the world perfectly using just three lines.

Learning Objectives

1.State the Pythagorean Theorem formula $a^2 + b^2 = c^2$
2.Calculate the length of the hypotenuse when given two legs
3.Identify right triangles using the converse of the Pythagorean Theorem

The Anatomy of a Right Triangle

Before we use the formula, we must identify the parts of our triangle. A right triangle is any triangle with exactly one $90^{\circ}$ angle (the 'square' corner). The two sides that form this corner are called the legs, usually labeled $a$ and $b$ . The side directly across from the right angle is the hypotenuse, labeled $c$ . The hypotenuse is always the longest side of the triangle. Think of it as the 'bridge' connecting the two legs. Understanding this relationship is the foundation of trigonometry and navigation.

Quick Check

In a right triangle, where is the hypotenuse located in relation to the right angle?

Look for the side that doesn't touch the square corner symbol.

Answer

The hypotenuse is always directly opposite the right angle and is the longest side.

The Magic Formula: $a^2 + b^2 = c^2$

The Pythagorean Theorem states that in any right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse. Mathematically, we write this as:

a^2 + b^2 = c^2

This means if you drew a literal square off each side of the triangle, the area of the two smaller squares added together would perfectly match the area of the largest square on the hypotenuse. This relationship allows us to find a missing side length as long as we know the other two.

Find the hypotenuse ( $c$ ) if the legs are $a = 3$ and $b = 4$ .

1. Write the formula: $a^2 + b^2 = c^2$ 2. Plug in the numbers: $3^2 + 4^2 = c^2$ 3. Square the numbers: $9 + 16 = c^2$ 4. Add them up: $25 = c^2$ 5. Find the square root: $\sqrt{25} = c$ 6. Final Answer: $c = 5$

Solving for the Hypotenuse

When you are looking for the hypotenuse, you are essentially solving for

c

. Because the formula gives you

c^2

, you must always perform the inverse operation of squaring, which is the square root. The isolated formula for the hypotenuse looks like this:

c = \sqrt{a^2 + b^2}

This is particularly useful in real-world scenarios, like finding the length of a ladder needed to reach a high window or calculating the diagonal screen size of a smartphone.

A ladder is placed $5$ feet away from a wall. It reaches a window that is $12$ feet high. How long is the ladder?

1. Identify the legs: $a = 5$ , $b = 12$ . 2. Set up the equation: $5^2 + 12^2 = c^2$ . 3. Calculate squares: $25 + 144 = c^2$ . 4. Sum the areas: $169 = c^2$ . 5. Take the square root: $c = \sqrt{169}$ . 6. Result: The ladder is $13$ feet long.

Quick Check

If $a^2 + b^2 = 100$ , what is the length of the hypotenuse $c$ ?

Take the square root of 100.

Answer

$c = 10$

The Converse: Is it a Right Triangle?

The Converse of the Pythagorean Theorem works backward. It says: If $a^2 + b^2 = c^2$ is true for a triangle's side lengths, then that triangle must be a right triangle. If the sides don't fit the formula, the triangle is either acute or obtuse. This is a powerful tool for construction workers and engineers to ensure that corners are perfectly 'square' without using a protractor.

A triangle has sides of $7$ , $24$ , and $25$ . Is it a right triangle?

1. Identify the longest side as $c$ : $c = 25$ . 2. Set up the check: $7^2 + 24^2 \stackrel{?}{=} 25^2$ . 3. Calculate: $49 + 576 \stackrel{?}{=} 625$ . 4. Add: $625 = 625$ . 5. Conclusion: Since the equation is balanced, it IS a right triangle.

Key Takeaways

1The Pythagorean Theorem only applies to right triangles.
2The formula is $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse.
3The hypotenuse is always the longest side and is opposite the right angle.
4Use the square root to find the actual side length after adding the squares.

Quick Check

Multiple Choice

What is the length of the hypotenuse if the legs are $6$ and $8$ ?

Multiple Choice

Which side of a right triangle is represented by ' $c$ ' in the formula?

True/False

A triangle with side lengths $5, 6, 7$ is a right triangle.

What's Next?

Review Tomorrow

In 24 hours, try to draw a right triangle from memory, label the legs and hypotenuse, and write the formula without looking at your notes.

Practice Activity

Look around your room for rectangular objects (like a phone or a book). Measure the width and height, then use the theorem to predict the diagonal length. Measure the diagonal to see how close you were!

Browse

Browse

The Pythagorean Theorem

Learning Objectives

The Anatomy of a Right Triangle

The Magic Formula: $a^2 + b^2 = c^2$

Solving for the Hypotenuse

The Converse: Is it a Right Triangle?

Key Takeaways

Quick Check

What's Next?

The Pythagorean Theorem

Learning Objectives

The Anatomy of a Right Triangle

The Magic Formula: $a^2 + b^2 = c^2$

Solving for the Hypotenuse

The Converse: Is it a Right Triangle?

Key Takeaways

Quick Check

What's Next?