Virginia SOL Math / Grade 8 / Measurement & Geometry

8.MG.4

The Pythagorean Theorem

What you'll learn

  • Verify and apply the Pythagorean Theorem
  • Find a missing side length of a right triangle
  • Solve practical problems using the theorem

Words to know

Right triangle —
a triangle with one 90 degree angle
Legs —
the two sides that form the right angle
Hypotenuse —
the side opposite the right angle; the longest side
Pythagorean Theorem —
a squared plus b squared equals c squared for a right triangle
Square root —
the inverse of squaring a number

Why it matters The Pythagorean Theorem lets us find distances we cannot measure directly, like the length of a ramp, ladder, or diagonal.

Worked examples

Model ItI do

A right triangle has legs 6 and 8. Find the hypotenuse.

Use a squared plus b squared equals c squared. So 6 squared plus 8 squared = 36 + 64 = 100. The hypotenuse is the square root of 100, which is 10.

Try It TogetherWe do

A right triangle has one leg 5 and hypotenuse 13. Find the other leg.

Rearrange: c squared minus a squared equals b squared. So 13 squared minus 5 squared = 169 - 25 = 144. The other leg is the square root of 144, which is 12.

Now YouYou do

A right triangle has legs 9 and 12. Find the hypotenuse.

Add the squares of the legs, then take the square root. Check your answer in the back of this chapter.

Practice

Mastered 0 of 11

Warm-up

Ease in with the basics.

  1. 1.A right triangle has legs 3 and 4. Find the hypotenuse.

  2. 2.What is the longest side of a right triangle called?

  3. 3.A right triangle has legs 6 and 8. Find the hypotenuse.

On Level

The core Quick Check from the book.

  1. 1.Legs 3 and 4. Find the hypotenuse.

  2. 2.Leg 8 and hypotenuse 17. Find the other leg.

  3. 3.Legs 5 and 12. Find the hypotenuse.

  4. 4.Is a triangle with sides 6, 8, 10 a right triangle?

  5. 5.A ladder reaches 12 feet up a wall with its base 5 feet out. How long is the ladder?

Challenge

Stretch your thinking.

  1. 1.A rectangular garden is 9 meters by 12 meters. How long is a straight diagonal path across it?

  2. 2.A right triangle has hypotenuse 26 and one leg 10. Find the other leg.

  3. 3.A triangle has sides 7, 24, and 25. Is it a right triangle? Explain.