Download 8-2 The Pythagorean Theorem and Its Converse 2-11

2/17/2016
Five-Minute Check (over Lesson 8–1)
CCSS
Then/Now
New Vocabulary
Theorem 8.4: Pythagorean Theorem
Proof: Pythagorean Theorem
Example 1: Find Missing Measures Using the Pythagorean
Theorem
Key Concept: Common Pythagorean Triples
Example 2: Use a Pythagorean Triple
Example 3: Standardized Test Example: Use the Pythagorean
Theorem
Theorem 8.5: Converse of the Pythagorean Theorem
Theorems: Pythagorean Inequality Theorems
Example 4: Classify Triangles
1
2/17/2016
Over Lesson 8–1
Find the geometric mean between 9 and 13.
Find the geometric mean between
Find the altitude a.
Find x, y, and z.
Over Lesson 8–1
Find the geometric mean between 9 and 13.
A. 2
B. 4
C.
D.
2
2/17/2016
Over Lesson 8–1
Find the geometric mean between
A.
B.
C.
D.
Over Lesson 8–1
Find the altitude a.
A. 4
B.
C. 6
D.
3
2/17/2016
Over Lesson 8–1
Find x, y, and z to the
nearest tenth.
A. x = 6, y = 8, z = 12
B. x = 7, y = 8.5, z = 15
C. x = 8, y ≈ 8.9, z ≈ 17.9
D. x = 9, y ≈ 10.1, z = 23
Over Lesson 8–1
Which is the best estimate for m?
A. 9
B. 10.8
C. 12.3
D. 13
4
2/17/2016
Content Standards
G.SRT.8 Use trigonometric ratios and the Pythagorean
Theorem to solve right triangles in applied problems.
G.MG.3 Apply geometric methods to solve problems
(e.g., designing an object or structure to satisfy
physical constraints or minimize cost; working with
typographic grid systems based on ratios).
Mathematical Practices
1 Make sense of problems and persevere in solving
them.
4 Model with mathematics.
You used the Pythagorean Theorem to
develop the Distance Formula.
• Use the Pythagorean Theorem.
• Use the Converse of the Pythagorean
Theorem.
5
2/17/2016
• Pythagorean triple
6
2/17/2016
Find Missing Measures Using the Pythagorean
Theorem
A. Find x.
Eureka!
The side opposite the right angle is the hypotenuse.
(leg)2 + (leg)2 = (hypotenuse)2
42 + 72 = x2
65 = x2
= 65
Find Missing Measures Using the Pythagorean
Theorem
B. Find x.
The hypotenuse is 12
(leg)2 + (leg)2= (hypotenuse)2
x2 + 82 = 122
x2 + 64 = 144
x2 = 80
7
2/17/2016
A. Find x.
A.
B.
C.
D.
B. Find x.
A.
B.
C.
D.
8
2/17/2016
To build your own Pythagorean triplet:
If m > n and both m & n are natural numbers then
leg = m2 – n2 , leg = 2mn and hypotenuse = m2 + n2
Use a Pythagorean Triple
Use a Pythagorean triple to find x. Explain your
reasoning.
Notice:
2 x 13 =
So:
= 2 x 5 = 10
2 x 12 =
Recall: 5,12,13 is a Pythagorean triplet
Answer:
x = 10
Check:
242 + 102 = 262
?
676 = 676
9
2/17/2016
Use a Pythagorean triple to find x.
A. 10
B. 15
C. 18
D. 24
Use the Pythagorean Theorem
A 20-foot ladder is placed against a building to
reach a window that is 16 feet above the ground.
How many feet away from the building is the
bottom of the ladder?
A3
B4
Method 1
C 12
D 15
Use a Pythagorean triple.
Notice:
So x = 3 x 4 = 12
5x4=
=4x4
This is a 3,4,5 ∆ with a scalar of 4
= 3 x 4 = 12
10
2/17/2016
Use the Pythagorean Theorem
A 20-foot ladder is placed against a building to
reach a window that is 16 feet above the ground.
How many feet away from the building is the
bottom of the ladder?
A3
B4
Method 2
C 12
D 15
Use Pythagorean theorem
+ + 16
= = 20 = 20
− 16
= 400 − 256
= 144 = ±12
= 12
A 10-foot ladder is placed against a building. The
base of the ladder is 6 feet from the building. How
high does the ladder reach on the building?
A. 6 ft
B. 8 ft
C. 9 ft
D. 10 ft
11
2/17/2016
12
2/17/2016
Classify Triangles
A. Determine whether 9, 12, and 15 can be the
measures of the sides of a triangle. If so, classify
the triangle as acute, right, or obtuse. Justify your
answer.
Step 1
Determine whether the measures can form a
triangle using the Triangle Inequality
Theorem.
9 + 12 > 15 9 + 15 > 12 12 + 15 > 9 The side lengths 9, 12, and 15 can form a
triangle.
Classify Triangles
A. Determine whether 9, 12, and 15 can be the
measures of the sides of a triangle. If so, classify
the triangle as acute, right, or obtuse. Justify your
answer.
Step 2
9
Classify the triangle
+ ? + 12 ? 15
81 + 144 ? 225
225 = 225
Answer: Since + = , the
triangle is a right triangle.
13
2/17/2016
Classify Triangles
B. Determine whether 10, 11, and 13 can be the
measures of the sides of a triangle. If so, classify
the triangle as acute, right, or obtuse. Justify your
answer.
Step 1
Determine whether the measures can form a
triangle using the Triangle Inequality
Theorem.
10 + 11 > 13 10 + 13 > 11 11 + 13 > 10 The side lengths 10, 11, and 13 can form a
triangle.
Classify Triangles
B. Determine whether 10, 11, and 13 can be the
measures of the sides of a triangle. If so, classify
the triangle as acute, right, or obtuse. Justify your
answer.
Step 2
10
Classify the triangle.
+ ? + 11 ? 13
100 + 121 ? 169
221 > 169
Answer: Since + > , the
triangle is an acute triangle.
14
2/17/2016
A. Determine whether the set of numbers 7, 8, and
14 can be the measures of the sides of a triangle. If
so, classify the triangle as acute, right, or obtuse.
Justify your answer.
A. yes, acute
B. yes, obtuse
C. yes, right
D. not a triangle
B. Determine whether the set of numbers 14, 18,
and 33 can be the measures of the sides of a
triangle. If so, classify the triangle as acute, right,
or obtuse. Justify your answer.
A. yes, acute
B. yes, obtuse
C. yes, right
D. not a triangle
15
2/17/2016
16