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# Lesson 3-5 Menu - PowerPoint PPT Presentation

Five-Minute Check (over Lesson 3-4) Main Idea and Vocabulary Targeted TEKS Key Concept: Pythagorean Theorem Example 1: Find the Length of a Side Example 2: Find the Length of a Side Key Concept: Converse of Pythagorean Theorem Example 3: Identify a Right Triangle. Lesson 3-5 Menu.

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Five-Minute Check(over Lesson 3-4)

Main Idea and Vocabulary

Targeted TEKS

Key Concept: Pythagorean Theorem

Example 1: Find the Length of a Side

Example 2: Find the Length of a Side

Key Concept: Converse of Pythagorean Theorem

Example 3: Identify a Right Triangle

• legs

• hypotenuse

• Pythagorean Theorem

• converse

Lesson 3-5 Ideas/Vocabulary

8.7The student uses geometry to model and describe the physical world. (C)Use pictures or models to demonstrate the Pythagorean Theorem. 8.9The student uses indirect measurement to solve problems.(A) Use the Pythagorean Theorem to solve real-life problems.

Lesson 3-5 TEKS

Write an equation to find the length of the missing side of the right triangle. Then find the missing length. Round to the nearest tenth, if necessary.

Lesson 3-5 Example 1

c = Definition of square root

Find the Length of a Side

c2 = a2 +b2 Pythagorean Theorem

c2 = 122 + 162 Replace a with 12 and b with 16.

c2 = 144 + 256 Evaluate 122 and 162.

c2 = 400 Add 144 and 256.

c = 20 or –20 Simplify.

Answer: The equation has two solutions, 20 and –20. However, the length of a side must be positive. So, the hypotenuse is 20 inches long.

Lesson 3-5 Example 1

Write an equation to find the length of the missing side of the right triangle. Then find the missing length. Round to the nearest tenth, if necessary.

A. 17 in.

B. 19 in.

C. 20 in.

D. 21 in.

Lesson 3-5 Example 1 CYP

= a Definition of square root

Find the Length of a Side

The hypotenuse of a right triangle is 33 centimeters long and one of its legs is 28 centimeters. What is a, the length of the other leg?

c2 = a2 +b2 Pythagorean Theorem

332 = a2 + 282 Replace c with 33 and b with 28.

1,089= a2 + 784 Evaluate 332 and 282.

1,089 – 784= a2 + 784 – 784 Subtract 784 from each side.

305 = a2 Simplify.

17.5 ≈ a Use a calculator.

Lesson 3-5 Example 2

Lesson 3-5 Example 2

The hypotenuse of a right triangle is 26 centimeters long and one of its legs is 17 centimeters. What is a, the length of the other leg?

Lesson 3-5 Example 2 CYP

252 = 72 + 242 Replace a with 7, b with 24, and c with 25.

?

625= 49 + 576 Evaluate 252, 72, and 242.

Identify a Right Triangle

The measures of three sides of a triangle are 24 inches, 7 inches, and 25 inches. Determine whether the triangle is a right triangle.

c2 = a2 +b2 Pythagorean Theorem

625= 625 Simplify.

Answer: The triangle is a right triangle.

Lesson 3-5 Example 3

• A

• B

• C

The measures of three sides of a triangle are 13 inches, 5 inches, and 12 inches. Determine whether the triangle is a right triangle.

A. It is a right triangle.

B. It is not a right triangle.

C. Not enough information to determine.

Lesson 3-5 CYP 3

Five-Minute Check(over Lesson 3-5)

Main Idea

Targeted TEKS

Example 1: Use the Pythagorean Theorem to Solve a Problem

Example 2: Test Example

8.7The student uses geometry to model and describe the physical world. (C)Use pictures or models to demonstrate the Pythagorean Theorem. 8.9The student uses indirect measurement to solve problems.(A) Use the Pythagorean Theorem to solve real-life problems.

Lesson 3-6 TEKS

Use the Pythagorean Theoremto Solve a Problem

RAMPS A ramp to a newly constructed building must be built according to the guidelines stated in the Americans with Disabilities Act. If the ramp is 24.1 feet long and the top of the ramp is 2 feet off the ground, how far is the bottom of the ramp from the base of the building?

Notice the problem involves a right triangle.

Use the Pythagorean Theorem.

Lesson 3-6 Example 1

= a Definition of square root

Use the Pythagorean Theoremto Solve a Problem

24.12 = a2 + 22 Replace c with 24.1 and b with 2.

580.81= a2+ 4 Evaluate 24.12 and 22.

580.81 – 4 = a2 + 4 – 4 Subtract 4 from each side.

576.81 = a2 Simplify.

24.0 ≈ a Simplify.

Answer: The end of the ramp is about 24 feet from the base of the building.

Lesson 3-6 Example 1

RAMPS If a truck ramp is 32 feet long and the top of the ramp is 10 feet off the ground, how far is the end of the ramp from the truck?

Lesson 3-6 Example 1 CYP

The cross-section of a camping tent is shown below. Find the width of the base of the tent.

A. 6 ft

B. 8 ft

C. 10 ft

D. 12 ft

Lesson 3-6 Example 2

From the diagram, you know that the tent forms two congruent right triangles. Let a represent half the base of the tent. Then w = 2a.

Lesson 3-6 Example 2

= a Definition of square root

Use the Pythagorean Theorem

Solve the Test Item

Use the Pythagorean Theorem.

c2 = a2 + b2 Write the relationship.

102 = a2 + 82c = 10 and b = 8

100 = a2 + 64 Evaluate 102 and 82.

100 – 64 = a2 + 64 – 64 Subtract 64 from each side.

36 = a2 Simplify.

6 = a Simplify.

Lesson 3-6 Example 2

The cross-section of a camping tent is shown below. Find the width of the base of the tent.

A. 6 ft

B. 8 ft

C. 10 ft

D. 12 ft

Answer:The width of the base of the tent is 2a or (2)6 = 12 feet. Therefore, choice D is correct.

Lesson 3-6 Example 2

This picture shows the cross-section of a roof. How long is each rafter, r?

A. 15 ft

B. 18 ft

C. 20 ft

D. 22 ft

Lesson 3-6 Example 2 CYP

Five-Minute Check(over Lesson 3-6)

Main Ideas and Vocabulary

Targeted TEKS

Example 1: Name an Ordered Pair

Example 2: Name an Ordered Pair

Example 3: Graphing Ordered Pairs

Example 4: Graphing Ordered Pairs

Example 5: Find Distance on the Coordinate Plane

Example 6: Use a Coordinate Plane to Solve a Problem

• Find the distance between two points on the coordinate plane.

• coordinate plane

• ordered pair

• x-coordinate

• abscissa

• y-coordinate

• ordinate

• origin

• y-axis

• x-axis

Lesson 3-7 Ideas/Vocabulary

8.7The student uses geometry to model and describe the physical world. (D)Locate and name points on the coordinate plane using ordered pairs of rational numbers. 8.9The student uses indirect measurement to solve problems.(A) Use the Pythagorean Theorem to solve real-life problems. Also addresses TEKS 8.1(C).

Lesson 3-7 TEKS

Answer: So, the ordered pair for point A is

Name an Ordered Pair

Name the ordered pair for point A.

• Start at the origin.

• Move right to find the x-coordinate of point A, which is 2.

Lesson 3-7 Example 1

Name the ordered pair for point A.

A.

B.

C.

D.

Lesson 3-7 Example 1 CYP

Answer: So, the ordered pair for point B is

Name an Ordered Pair

Name the ordered pair for point B.

• Start at the origin.

• Move down to find the y-coordinate, which is –2.

Lesson 3-7 Example 2

Name the ordered pair for point B.

A.

B.

C.

D.

Lesson 3-7 Example 2 CYP

Graph and label point J(–3, 2.75).

• Start at the origin and move 3 units to the left. Then move up 2.75 units.

• Draw a dot and label it J(–3, 2.75).

Lesson 3-7 Example 3

B.

C.

D.

• A

• B

• C

• D

Graph and label point J(–2.5, 3.5).

Lesson 3-7 Example 3 CYP

• Start at the origin and move 4 units to the right. Then move down units.

• Draw a dot and label it K

Graphing Ordered Pairs

Lesson 3-7 Example 4

A.

B.

C.

D.

• A

• B

• C

• D

Lesson 3-7 Example 4 CYP

Graph the ordered pairs (0, –6) and (5, –1). Then find the distance between the points.

Let c = the distance between the two points, a = 5, and b = 5.

Lesson 3-7 Example 5

= Definition of square root

Find Distance in the Coordinate Plane

c2 = a2 + b2 Pythagorean Theorem

c2 = 52 + 52 Replace a with 5 and b with 5.

c2 = 50 52 + 52 = 50

c≈ 7.1 Simplify.

Lesson 3-7 Example 5

Graph the ordered pairs (0, –3) and (2, –6). Then find the distance between the points.

Lesson 3-7 Example 5 CYP

Use a Coordinate Plane to Solve a Problem

TRAVEL Melissa lives in Chicago, Illinois. A unit on the grid of her map shown below is 0.08 mile. Find the distance between McCormickville at (–2, –1) and Lake Shore Park at (2, 2).

Let c = the distance between McCormickville and Lake Shore Park. Then a = 3 and b = 4.

Lesson 3-7 Example 6

= Definition of square root

Use a Coordinate Plane to Solve a Problem

c2 = a2 + b2 Pythagorean Theorem

c2 = 32 + 42 Replace a with 3 and b with 4.

c2 = 25 32 + 42 = 25

c= 5 Simplify.

The distance between McCormickville and Lake Shore Park is 5 units on the map.

Answer:Since each unit equals 0.08 mile, the distance is 0.08  5 or 0.4 mile.

Lesson 3-7 Example 6

TRAVEL Sato lives in Chicago. A unit on the grid of his map shown below is 0.08 mile. Find the distance between Shantytown at (2, –1) and the intersection of N. Wabash Ave. and E. Superior St. at (–3, 1).