Pre-Algebra
6-3 The Pythagorean Theorem
6-3 The Pythagorean Theorem
Pre-Algebra
Warm Up
Problem of the Day
Lesson Presentation
Pre-Algebra
6-3 The Pythagorean Theorem
Warm UpGraph the figures with the given verticals and find the area.
1. (–1, –1), (–1, 3), (6, –1)
2. (2, 1), (8, 1), (6, –3)
3. (3, –2), (15, –2), (14, 6), (4, 6)
Pre-Algebra
6-3 The Pythagorean Theorem
14 units2
12 units2
88 units2
Pre-Algebra
6-3 The Pythagorean Theorem
Problem of the DayA side of a square A is 5 times the length of square B. How many times as great is the area of square A than the area of square B? 25
Pre-Algebra
6-3 The Pythagorean Theorem
Learn to use the Pythagorean Theorem and its converse to solve problems.
6-3 The Pythagorean Theorem
Pre-Algebra
6-3 The Pythagorean Theorem
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.
Additional Example 1A: Find the the Length of a Hypotenuse
Find the length of the hypotenuse.
41 = c2
Pre-Algebra
6-3 The Pythagorean Theorem
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.
Additional 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)
Pre-Algebra
6-3 The Pythagorean Theorem
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.
Pre-Algebra
6-3 The Pythagorean Theorem
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.
Pre-Algebra
6-3 The Pythagorean Theorem
Additional 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.
Pre-Algebra
6-3 The Pythagorean Theorem
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.
Pre-Algebra
6-3 The Pythagorean Theorem
Additional 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.
Pre-Algebra
6-3 The Pythagorean Theorem
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
Pre-Algebra
6-3 The Pythagorean Theorem
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.
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