Congruent and Similar Solids LESSON 12–8 Congruent and Similar Solids

Five-Minute Check (over Lesson 12–7) TEKS Then/Now New Vocabulary Key Concept: Similar Solids Key Concept: 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 Lesson Menu

Name a line not containing point P on the sphere. B. C. D. 5-Minute Check 1

Name a triangle in the sphere. A. ΔVQS B. ΔRTU C. ΔPQR D. ΔPXW 5-Minute Check 2

Name a segment containing point Q in the sphere. B. C. D. TU 5-Minute Check 3

A. Yes, through 2 points there is exactly one line. 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. 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. 5-Minute Check 4

Which of the following is represented by a line in spherical geometry? A. triangle B. great circle C. radius D. diameter 5-Minute Check 5

Mathematical Processes G.1(E), G.1(F) Targeted TEKS G.11(D) Apply the formulas for the volume of three-dimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure. Mathematical Processes G.1(E), G.1(F) TEKS

You compared surface areas and volumes of spheres. Identify congruent or similar solids. Use properties of similar solids. Then/Now

similar solids congruent solids Vocabulary

Concept

Concept

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. Substitution Simplify. Example 1

Substitution Simplify. Substitution Simplify. Identify Similar and Congruent Solids Substitution Simplify. Substitution Simplify. 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. Example 1

Compare the ratios between the corresponding parts of the cones. 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. Example 1

Substitution Simplify. Substitution Identify Similar and Congruent Solids Substitution Simplify. Substitution Answer: Since the ratios are not the same, the cones are neither similar nor congruent. Example 1

A. Determine whether the pair of solids is similar, congruent, or neither. A. similar B. congruent C. neither Example 1

B. Determine whether the pair of solids is similar, congruent, or neither. A. similar B. congruent C. neither Example 1

Concept

First, find the scale factor. 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. Example 2

If the scale factor is , then the ratio of the volumes is . 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. Example 2

Two similar cones have radii of 5 inches and 15 inches 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? A. 1:3 B. 1:9 C. 1:27 D. 1:81 Example 2

Analyze You know the volume of the softballs. 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. Analyze You know the volume of the softballs. Formulate Use Theorem 12.1 to write a ratio comparing the volumes. Then find the scale factor and use it to find r. Example 3

Write a ratio comparing volumes. Use Similar Solids to Find Unknown Values Determine Write a ratio comparing volumes. = Simplify. ≈ Write as . Example 3

Find the cross products. Use Similar Solids to Find Unknown Values Ratio of radii Scale factor Find the cross products. r ≈ 1.45 Solve for r. Answer: So, the radius of the smaller softball is about 1.45 inches. Example 3

Use Similar Solids to Find Unknown Values Justify  Evaluate For the two spheres to be similar, all corresponding measurements must have the same scale factor. Example 3

CONTAINERS The containers below are similar cylinders CONTAINERS The containers below are similar cylinders. Find the height h of the smaller container. A. 2 in. B. 3 in. C. 4 in. D. 5 in. Example 3

Congruent and Similar Solids LESSON 12–8 Congruent and Similar Solids