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anthony-russell
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About 2,500 years ago, a Greek mathematician named Pythagorus discovered a special relationship between the sides of right triangles.
a2 + b2 = c2
Used to find a missing side of a right triangle
a & b always shortest sides * c is always longest side
Steps
1. Identify what sides you have and which side you are looking for.
2. Substitute the values you have into the appropriate places in the Pythagorean Theorem a2 + b2 = c2
3. Do your squaring first… then solve the 2-Step equation.
TOTD: if your answer under the radical is not a perfect square, leave your answer under the radical.
4
5
c
6.40 c
A.
Pythagorean TheoremSubstitute for a and b.
a2 + b2 = c2
42 + 52 = c2
16 + 25 = c2
41 = c
Simplify powers. Solve for c; c = c2.
Example 1A: Find the Length of a Hypotenuse
Find the length of the hypotenuse.
41 = c2
Example: 2 Finding the Length of a Leg in a Right Triangle
25
7
b
576 = 24b = 24
a2 + b2 = c2
72 + b2 = 252
49 + b2 = 625–49 –49
b2 = 576
Solve for the unknown side in the right triangle.
Pythagorean TheoremSubstitute for a and c. Simplify powers.
Try This: Example 1A
5
7
cA.
Find the length of the hypotenuse.
8.60 c
Pythagorean TheoremSubstitute for a and b.
a2 + b2 = c2
52 + 72 = c2
25 + 49 = c2
74 = cSimplify powers. Solve for c; c = c2.
Try This: Example 2
b 11.31
12
4
ba2 + b2 = c2
42 + b2 = 122
16 + b2 = 144–16 –16
b2 = 128
128 11.31
Solve for the unknown side in the right triangle.
Pythagorean TheoremSubstitute for a and c. Simplify powers.
15 = c
B.
Pythagorean TheoremSubstitute for a and b.
a2 + b2 = c2
92 + 122 = c2
81 + 141 = c2
225 = cSimplify powers. Solve for c; c = c2.
Example 1B: Find the the Length of a Hypotenuse
Find the length of the hypotenuse.
triangle with coordinates
(1, –2), (1, 7), and (13, –2)
B. triangle with coordinates (–2, –2), (–2, 4), and (3, –2)
x
y
The points form a right triangle.
(–2, –2)
(–2, 4)
(3, –2)
Try This: Example 1B
Find the length of the hypotenuse.
7.81 c
Pythagorean Theorema2 + b2 = c2
62 + 52 = c2
36 + 25 = c2
61 = cSimplify powers. Solve for c; c = c2.
Substitute for a and b.
Example 3: Using the Pythagorean Theorem to Find Area
a6 6
4 4
a2 + b2 = c2
a2 + 42 = 62
a2 + 16 = 36
a2 = 20a = 20 units ≈ 4.47 units
Find the square root of both sides.
Substitute for b and c.Pythagorean Theorem
A = hb = (8)( 20) = 4 20 units2 17.89 units212
12
Use the Pythagorean Theorem to find the height of the triangle. Then use the height to find the area of the triangle.
a2 + b2 = c2
a2 + 22 = 52
a2 + 4 = 25
a2 = 21
a = 21 units ≈ 4.58 units
Find the square root of both sides.
Substitute for b and c.
Pythagorean Theorem
A = hb = (4)( 21) = 2 21 units2 4.58 units212
12
Try This: Example 3
Use the Pythagorean Theorem to find the height of the triangle. Then use the height to find the area of the triangle.
a5 5
2 2
Lesson Quiz
1. Find the height of the triangle.
2. Find the length of side c to the nearest meter.
3. Find the area of the largest triangle.
4. One leg of a right triangle is 48 units long, and the hypotenuse is 50 units long. How long is the other leg?
8m
12m
60m2
14 units
h
c10 m
6 m 9 m
Use the figure for Problems 1-3.