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

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Presentation transcript:

ESSENTIAL OBJECTIVE Use the Pythagorean Theorem and the Distance Formula.

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

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.

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.

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 = 52 + 122 c2 = 25 + 144 c2 = 169 c2 = 169 c = 13 ANSWER The length of the hypotenuse is 13. 5

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

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

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.

WE DO….! 2.

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

YOU DO….! 3.

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

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 = 32 + 42 (AB)2 = 9 + 16 (AB)2 = 25 the positive square root. (AB)2 = 25 AB = 5 13

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

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

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

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

Checkpoint Use the Distance Formula 5.

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

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

HW Practice 4.4A