1. The Pythagorean Theorem
Lesson 3.3.1

2. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
California Standard:
Measurement and Geometry 3.3
Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement.
What it means for you:
You’ll learn about an equation that you can use to find a missing side length of a right triangle.
Key words:
Pythagorean theorem
right triangle
hypotenuse
legs
right angle

3. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
You will have come across right triangles before — they’re just triangles that have one corner that’s a 90° angle.
Well, there’s a special formula that links the side lengths of a right triangle — it comes from the Pythagorean theorem.

4. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
The Pythagorean Theorem is About Right Triangles
A right triangle is any triangle that has a 90° angle (or right angle) as one of it corners. c b a
You need to know the names of the parts of a right triangle:
The hypotenuse is the longest side of the triangle. It’s the side directly opposite the right angle.
90°
Right angle
The other two sides of the triangle are called the legs.

5. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
In diagrams of right triangles, the hypotenuse is usually labeled as c, and the two legs as a and b. It doesn’t matter which leg you label a, and which you label b.
Leg = b
Leg = a
Hypotenuse = c

6. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
Guided Practice
Complete the missing labels on the diagram. 1.
Hypotenuse 2.
Right angle
90° 3.
Legs
Solution follows…

7. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
Guided Practice
In Exercises 4–7 say which side of the right triangle is the hypotenuse. I II I II I II
III
III I II I II II
III
III
III
Solution follows…

8. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
The Theorem Links Side Lengths of Right Triangles
Pythagoras was a Greek mathematician who lived around 500 B.C. A famous theorem about right triangles is named after him. It’s called the Pythagorean theorem:
For any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs.
This all sounds very complicated, but it’s not so bad once you know what it actually means.

9. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
Look again at the right triangle. Now add three squares whose side lengths are the same as the side lengths of the triangle:
Area = c2
Area = b2 c b a
Area = a2

10. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
What the Pythagorean theorem is saying is that the area of the red square is the same as the area of the blue square plus the area of the green square.
Area = c2 c2 b2 a2 + =
Area = b2 c b a
Area = a2

11. For any right triangle: c2 = a2 + b2
Lesson
3.3.1
The Pythagorean Theorem
So this is what the Pythagorean theorem looks like written algebraically:
For any right triangle: c2 = a2 + b2
It means that if you know the lengths of two sides of a right triangle, you can always find the length of the other side using the equation.

12. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
You Can Check the Theorem Using a Right Triangle
You can check for yourself that the theorem works by measuring the side lengths of right triangles, and putting the values into the equation.
Don’t forget — the Pythagorean theorem only works on right triangles. It won’t work on any other type.

13. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
Example 1
Use the right triangle shown to verify the Pythagorean theorem.
5 units
4 units
Solution
3 units
Read the side lengths from the diagram
a = 3 units, b = 4 units, c = 5 units
c2 = a2 + b2
Write out the formula
52 =
Substitute the values
25 =
Simplify the equation
25 = 25
Solution follows…

14. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
Guided Practice
Use the right triangles in Exercises 8–9 to verify the Pythagorean theorem.
13 units
15 units
12 units
12 units
5 units
132 = = = 169
9 units
152 = = = 225
Solution follows…

15. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
Guided Practice
Use the right triangles in Exercises 10–11 to verify the Pythagorean theorem.
2.5 units
0.7 units
2.4 units
4.1 units
4.12 = = = 16.81
2.52 = = = 6.25
4 units
0.9 units
Solution follows…

16. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
Independent Practice
In Exercises 1–3 say whether the triangle is a right triangle or not.
Yes
Yes No
Solution follows…

17. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
Independent Practice
In Exercises 4–6 say which side of the right triangle is the hypotenuse. 6. II I II
III II
III I I I
III II II
Solution follows…

18. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
Independent Practice
Use the triangles in Exercises 7–8 to verify the Pythagorean theorem.
10 units
8 units
17 units
15 units
6 units
102 = = = 100
172 = = = 289
8 units
Solution follows…

19. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
Independent Practice
Use the triangles in Exercises 9–10 to verify the Pythagorean theorem.
2 units
1.25 units
1 unit
1.6 units
0.75 units
1.2 units
= = =
22 = = = 4
Solution follows…

20. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
Independent Practice
11. Victor used the triangle shown on the right to try to verify the Pythagorean theorem. Explain why his work is wrong.
Victor’s work:
92 = = = 369
15 units
12 units
Victor has not correctly identified the hypotenuse. The theorem says c2 = a2 + b2 where c is the longest side, or hypotenuse. The first line of his work should be 152 =
9 units
Solution follows…

21. The Pythagorean Theorem
Lesson
3.3.1
The Pythagorean Theorem
Round Up
The Pythagorean theorem describes the relationship between the lengths of the hypotenuse and the legs of a right triangle.
It means that when you know the lengths of two of the sides of a right triangle, you can always find the length of the third side.
You’ll get a lot of practice at using the Pythagorean theorem in the next few Lessons.