Five-Minute Check (over Lesson 12–7)

Then/Now

New Vocabulary

Key Concepts: Similar Solids

Key Concepts: Congruent Solids

Example 1:Identify Similar and Congruent Solids

Theorem 12.1

Example 2:Use Similar Solids to Write Ratios

Example 3:Real-World Example: Use Similar Solids to Find Unknown Values

Over Lesson 12–7

Name a line not containing point P on the sphere.

A.

B.

C.

D.

Over Lesson 12–7

A. ΔVQS

B. ΔRTU

C. ΔPQR

D. ΔPXW

Name a triangle in the sphere.

Over Lesson 12–7

Name a segment containing point Q in the sphere.

A.

B.

C.

D. TU

Over Lesson 12–7

A. Yes, through 2 points there is exactly one line.

B. Yes, the points on any great circle or arc of a great circle can be put into one to one correspondence with real numbers.

C. No, AC may not be the distance from A to C through B. It may be the distance the other direction around the sphere.

Tell whether the following statement from Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain why.If B is between A and C, then AB + BC = AC.

Over Lesson 12–7

A. triangle

B. great circle

C. radius

D. diameter

Which of the following is represented by a line in spherical geometry?

You compared surface areas and volumes of spheres.

• Identify congruent or similar solids.

• Use properties of similar solids.

• similar solids

• congruent solids

Identify Similar and Congruent Solids

A. Determine whether the pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor.

Find the ratios between the corresponding parts of the square pyramids.

Simplify.

Substitution

Identify Similar and Congruent Solids

Answer: The ratios of the measures are equal, so we can conclude that the pyramids are similar. Since the scale factor is not 1, the solids are not congruent.

Substitution

Simplify.

Simplify.

Substitution

Identify Similar and Congruent Solids

B. Determine whether the pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor.

Compare the ratios between the corresponding parts of the cones.

Identify Similar and Congruent Solids

Answer: Since the ratios are not the same, the cones are neither similar nor congruent.

Simplify.

Substitution

Substitution

A. similar

B. congruent

C. neither

A. Determine whether the pair of solids is similar, congruent, or neither.

A. similar

B. congruent

C. neither

B. Determine whether the pair of solids is similar, congruent, or neither.

Use Similar Solids to Write Ratios

Two similar cones have radii of 9 inches and 12 inches. What is the ratio of the volume of the smaller cone to the volume of the larger cone?

First, find the scale factor.

Write a ratio comparing the radii.

Use Similar Solids to Write Ratios

If the scale factor is , then the

ratio of the volumes is .

Answer: So, the ratio of the volume is 27:64.

A. 1:3

B. 1:9

C. 1:27

D. 1:81

Two similar cones have radii of 5 inches and 15 inches. What is the ratio of the volume of the smaller cone to the volume of the larger cone?

Use Similar Solids to Find Unknown Values

SOFTBALLS The softballs shown are similar spheres. Find the radius of the smaller softball if the radius of the larger one is about 1.9 cubic inches.

Understand You know the volume of the softballs.

Plan Use Theorem 12.1 to write a ratio comparing the volumes. Then find the scale factor and use it to find r.

Use Similar Solids to Find Unknown Values

Solve

Write a ratio comparing volumes.

Simplify.

Write as .

=

≈

Use Similar Solids to Find Unknown Values

Ratio of radii Scale factor

Find the cross products.

Solve for r.

Answer: So, the radius of the smaller softball is about 1.45 inches.

r ≈ 1.45

Use Similar Solids to Find Unknown Values

Check

A. 2 in.

B. 3 in.

C. 4 in.

D. 5 in.

CONTAINERS The containers below are similar cylinders. Find the height h of the smaller container.