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Applying the Pythagorean Theorem and Its Converse4-9
Warm UpWarm Up
Lesson PresentationLesson Presentation
Problem of the DayProblem of the Day
Lesson QuizzesLesson Quizzes
Applying the Pythagorean Theorem and Its Converse4-9
Warm UpFind the distance between each pair of points.
1. (8, 2) and (8, 7)
2. (–2, 4) and (5, 4)
3. (–1, –1) and (9, –1)
4. (–8, –4) and (–2, –4)
5 units
7 units
10 units
6 units
Applying the Pythagorean Theorem and Its Converse4-9
Problem of the DayThe sum of the squares of two positive numbers is 100. One number is two more than the other. What are the numbers?
6 and 8
Applying the Pythagorean Theorem and Its Converse4-9
Learn to use the Distance Formula and the Pythagorean Theorem and its converse to solve problems.
Applying the Pythagorean Theorem and Its Converse4-9
7.615 c
Use the Pythagorean TheoremSimplify.
72 + 32 = c2
49 + 9 = c2
58 = c2
58 = c
Add.
Additional Example 1: Marketing Application
What is the diagonal length of the projector screen?
The diagonal length should be given as about 7.62 feet.
Applying the Pythagorean Theorem and Its Converse4-9
14.42 c
Use the Pythagorean TheoremSimplify.
122 + 82 = c2
144 + 64 = c2
208 = c2
208 = c
Add.
Check It Out: Example 1
What is the diagonal length of the projector screen?
The diagonal length should be given as about 14.4 feet.
12 ft
8 ft
Applying the Pythagorean Theorem and Its Converse4-9
Applying the Pythagorean Theorem and Its Converse4-9
Use the Distance Formula.
J and K
Additional Example 2A: Finding Distance on the Coordinate Plane
Find the distances between the points to the nearest tenth.
Let A be (x1, y1)² and B be (x2, y2)².
Applying the Pythagorean Theorem and Its Converse4-9
Substitute.
Subtract.
Additional Example 2A Continued
Find the distances between the points to the nearest tenth.
d = (0 – (–4))² + (–3 – 0 )²
d = (4)² + (–3)²
Simplify powers.
d = 16 + 9
The distance between A and B is 5 units.
Applying the Pythagorean Theorem and Its Converse4-9
L and M
Additional Example 2B: Finding Distance on the Coordinate Plane
Find the distances between the points to the nearest tenth.
Use the Distance Formula.
Let A be (x1, y1)² and B be (x2, y2)².
Applying the Pythagorean Theorem and Its Converse4-9
Substitute.
Subtract.
Additional Example 2B Continued
Find the distances between the points to the nearest tenth.
d = (5 – 4)² + (–3 – 0)²
d = (1)² + (–3)²
Simplify powers.
d = 1 + 9
The distance between L and M is about 3.2 units.
Applying the Pythagorean Theorem and Its Converse4-9
J and L
Check It Out: Example 2A
Find the distances between the points to the nearest tenth.
Use the Distance Formula.
Let A be (x1, y1)² and B be (x2, y2)².
Applying the Pythagorean Theorem and Its Converse4-9
Substitute.
Subtract.
Check It Out: Example 2A Continued
Find the distances between the points to the nearest tenth.
d = (4 – (–4))² + (0 – 0)²
d = (8)² + (0)²
Simplify powers.
d = 64 + 0
The distance between J and L is 8 units.
Applying the Pythagorean Theorem and Its Converse4-9
K and M
Check It Out: Example 2B
Find the distances between the points to the nearest tenth.
Use the Distance Formula.
Let A be (x1, y1)² and B be (x2, y2)².
Applying the Pythagorean Theorem and Its Converse4-9
Substitute.
Subtract.
Check It Out: Example 2B Continued
Find the distances between the points to the nearest tenth.
d = ((–4) – 0)² + (– 3 – (–3)²
d = (–4)² + (0)²
Simplify powers.
d = 16 + 0
The distance between K and M is 4 units.
Applying the Pythagorean Theorem and Its Converse4-9
Additional Example 3A: Identifying a Right Triangle
a2 + b2 = c2
92 + 122 = 152
81 + 144 = 225
225 = 225 √
Substitute.
Compare a² + b² to c².
The side lengths form a right triangle.
Tell whether the given side lengths form a right triangle.
9, 12, 15
Simplify.
Add.
Applying the Pythagorean Theorem and Its Converse4-9
Additional Example 3B: Identifying a Right Triangle
a2 + b2 = c2
82 + 102 = 132
63 + 100 = 169
163 ≠ 169 x
Substitute.
Compare a² + b² to c².
The side lengths do not form a right triangle.
Tell whether the given side lengths form a right triangle.
8, 10, 13
Simplify.
Add.
Applying the Pythagorean Theorem and Its Converse4-9
Check It Out: Example 3A
a2 + b2 = c2
52 + 62 = 92
25 + 36 = 81
61 ≠ 81 x
Substitute.
Compare a² + b² to c².
The side lengths do not form a right triangle.
Tell whether the given side lengths form a right triangle.
5, 6, 9
Simplify.
Add.
Applying the Pythagorean Theorem and Its Converse4-9
Check It Out: Example 3B
a2 + b2 = c2
82 + 152 = 172
64 + 225 = 289
289 = 289 √
Substitute.
Compare a² + b² to c².
The side lengths form a right triangle.
Tell whether the given side lengths form a right triangle.
8, 15, 17
Simplify.
Add.
Applying the Pythagorean Theorem and Its Converse4-9
Standard Lesson Quiz
Lesson Quizzes
Lesson Quiz for Student Response Systems
Applying the Pythagorean Theorem and Its Converse4-9
Lesson Quiz: Part I
1. width = 9 in., length = 40 in.
2. base length = 20 m, height = 15 m.
41 in.
25 m
Find the length of the diagonal of the rectangle.
Applying the Pythagorean Theorem and Its Converse4-9
Lesson Quiz: Part II
3. Q and R
4. S and T
≈ 6.3
≈ 2.2
Find the distance between the points to the nearest tenth.
Applying the Pythagorean Theorem and Its Converse4-9
Lesson Quiz: Part III
5. 12 cm, 13 cm, 16 cm
6. 11 ft, 60 ft, 61 ft
no
yes
Tell whether the given side lengths form a right triangle.
Applying the Pythagorean Theorem and Its Converse4-9
1. Find the length of the diagonal of the rectangle.
width = 5 m, length = 12 m
A. 8 m
B. 11 m
C. 13 m
D. 17 m
Lesson Quiz for Student Response Systems
Applying the Pythagorean Theorem and Its Converse4-9
2. Find the distance between the points to the nearest tenth.
Q and S
A. 6.0 m
B. 6.1 m
C. 7.0 m
D. 37.1 m
Lesson Quiz for Student Response Systems
Applying the Pythagorean Theorem and Its Converse4-9
3. Tell whether the given side lengths form a right triangle.
12 ft, 13 ft, 14 ft
A. yes
B. no
Lesson Quiz for Student Response Systems
