1. ESSENTIAL OBJECTIVE Use the Pythagorean Theorem and the Distance Formula.

2. Right Triangle
The sides that form the right angle are called the legs.
The side opposite the right angle is called the hypotenuse.

3. The Pythagorean Theorem
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

4. The Pythagorean Theorem
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

5. Find the length of the hypotenuse.
Example 1
Find the Length of the Hypotenuse
Find the length of the hypotenuse.
SOLUTION
(hypotenuse)2 = (leg)2 + (leg)2
c2 =
c2 =
c2 = 169
c2 =
169
c = 13
ANSWER
The length of the hypotenuse is 13. 5

6. Find the unknown side length.
Example 2
Find the Length of a Leg
Find the unknown side length.
SOLUTION
(hypotenuse)2 = (leg)2 + (leg)2
142 = 72 + b2
196 = 49 + b2
196 – 49 = 49 + b2 – 49
147 = b2
147 = b2
12.1 ≈ b
ANSWER
The side length is about 12.1. 6

7. Find the unknown side length.
I DO…..! 1.

8. I DO…..! Checkpoint Find the unknown side length. 1.
Find the Lengths of the Hypotenuse and Legs
Find the unknown side length.
I DO…..! 1.

9. WE DO….! 2.

10. Checkpoint
Find the Lengths of the Hypotenuse and Legs
WE DO….! 2.

11. YOU DO….! 3.

12. Checkpoint
Find the Lengths of the Hypotenuse and Legs
YOU DO….! 3.

13. Find the distance between the points A(1, 2) and B(4, 6).
Example 3
Find the Length of a Segment
Find the distance between the points
A(1, 2) and B(4, 6).
SOLUTION
Using the Pythagorean Theorem.
(hypotenuse)2 = (leg)2 + (leg)2
(AB)2 =
(AB)2 =
(AB)2 = 25
the positive square root.
(AB)2 = 25
AB = 5 13

14. The Distance Formula
If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the distance between A and B is
AB = (x2 - x1)2 + (y2 - y1)2

15. Find the distance between D(1, 2) and E(3, 2).
Example 4
Use the Distance Formula
Find the distance between D(1, 2) and E(3, 2).
SOLUTION
Begin by plotting the points in a coordinate plane.
x1 = 1, y1 = 2, x2 = 3, and y2 = –2.
The Distance Formula
DE =
(x2 – x1)2 + (y2 – y1)2
Substitute. =
(3 – 1)2 + (–2 – 2)2
Simplify. =
22 + (–4)2
Multiply. =
4 + 16
Add. = 20 ≈
4.5
Approximate with a calculator.
ANSWER
The distance between D and E is about 4.5 units. 15

16. Find the distance between D(1, 2) and E(3, 2).
Example 4
Use the Distance Formula
Find the distance between D(1, 2) and E(3, 2).
ANSWER
The distance between D and E is about 4.5 units. 16

17. Find the distance between the points.
Checkpoint
Use the Distance Formula
Find the distance between the points. 4.

18. Checkpoint
Use the Distance Formula 5.

19. REVIEW Find the value of x. Tell what theorem(s) you used. 1. ANSWER
x = 70; Base Angles Theorem 2.
ANSWER
x = 90; Base Angles Theorem, Triangle Sum Theorem

20. Find the value of x. 3.
ANSWER
x = 4 4.
ANSWER
x = 9

21. HW Practice 4.4A