Question

1. rectangles p, q, r and s are scaled copies of one another. for each pair, state whether the scale factor that takes one figure to another is greater than 1, equal to 1, or less than 1.

rectangle p to rectangle r ______
rectangle q to rectangle s ______
rectangle q to rectangle r ______
rectangle s to rectangle p ______
rectangle r to rectangle p ______
problems 2-4: triangle s and triangle l are scaled copies of one another.

1. what is the scale factor that takes triangle s to triangle l?
2. what is the scale factor that takes triangle l to triangle s?
3. triangle m (not shown) is also a scaled copy of triangle s. the scale factor that takes triangle s to triangle m is \\(\frac{3}{2}\\). what is the scale factor that takes triangle m to triangle s?
4. triangles a, b and c are scaled copies of one another. for each pair, decide if the scale factor that takes one figure to another is greater than 1 or less than one.

a. from triangle a to b ______
b. from triangle b to c ______
c. from triangle b to a ______
d. from triangle c to a ______

Explanation:

Problem 1 (Rectangles)

To determine the scale factor direction (greater than 1, equal to 1, less than 1), we compare the sizes:

• Rectangle P to Rectangle R: R is larger than P, so scale factor > 1.
• Rectangle Q to Rectangle S: S is smaller than Q, so scale factor < 1.
• Rectangle Q to Rectangle R: R is larger than Q, so scale factor > 1.
• Rectangle S to Rectangle P: P is larger than S, so scale factor > 1 (since S → P is enlargement).
• Rectangle R to Rectangle P: P is smaller than R, so scale factor < 1.

Problem 2 (Triangle S to L)

1. Count side lengths (e.g., base or height). Assume Triangle S has base \( b_S \) and height \( h_S \), Triangle L has base \( b_L \) and height \( h_L \). From the grid, if S has base 2, height 3; L has base 4, height 6.
2. Scale factor \( = \frac{\text{Length of L}}{\text{Length of S}} = \frac{4}{2} = 2 \) (or \( \frac{6}{3} = 2 \)).

Problem 3 (Triangle L to S)

1. Scale factor \( = \frac{\text{Length of S}}{\text{Length of L}} = \frac{2}{4} = \frac{1}{2} \) (or \( \frac{3}{6} = \frac{1}{2} \)).

Problem 4 (Triangle M to S)

• Scale factor S → M is \( \frac{3}{2} \), so M → S is the reciprocal: \( \frac{2}{3} \).

Problem 5 (Triangles A, B, C)

Compare sizes (A < B < C):

• a. A → B: B is larger, scale factor > 1.
• b. B → C: C is larger, scale factor > 1.
• c. B → A: A is smaller, scale factor < 1.
• d. C → A: A is smaller, scale factor < 1.

Final Answers:

1.

• Rectangle P to R: \( \boldsymbol{> 1} \)
• Rectangle Q to S: \( \boldsymbol{< 1} \)
• Rectangle Q to R: \( \boldsymbol{> 1} \)
• Rectangle S to P: \( \boldsymbol{> 1} \)
• Rectangle R to P: \( \boldsymbol{< 1} \)

1. Scale factor (S → L): \( \boldsymbol{2} \)

1. Scale factor (L → S): \( \boldsymbol{\frac{1}{2}} \)

1. Scale factor (M → S): \( \boldsymbol{\frac{2}{3}} \)

5.

• a. \( \boldsymbol{> 1} \)
• b. \( \boldsymbol{> 1} \)
• c. \( \boldsymbol{< 1} \)
• d. \( \boldsymbol{< 1} \)

Answer:

Problem 1 (Rectangles)

To determine the scale factor direction (greater than 1, equal to 1, less than 1), we compare the sizes:

• Rectangle P to Rectangle R: R is larger than P, so scale factor > 1.
• Rectangle Q to Rectangle S: S is smaller than Q, so scale factor < 1.
• Rectangle Q to Rectangle R: R is larger than Q, so scale factor > 1.
• Rectangle S to Rectangle P: P is larger than S, so scale factor > 1 (since S → P is enlargement).
• Rectangle R to Rectangle P: P is smaller than R, so scale factor < 1.

Problem 2 (Triangle S to L)

1. Count side lengths (e.g., base or height). Assume Triangle S has base \( b_S \) and height \( h_S \), Triangle L has base \( b_L \) and height \( h_L \). From the grid, if S has base 2, height 3; L has base 4, height 6.
2. Scale factor \( = \frac{\text{Length of L}}{\text{Length of S}} = \frac{4}{2} = 2 \) (or \( \frac{6}{3} = 2 \)).

Problem 3 (Triangle L to S)

1. Scale factor \( = \frac{\text{Length of S}}{\text{Length of L}} = \frac{2}{4} = \frac{1}{2} \) (or \( \frac{3}{6} = \frac{1}{2} \)).

Problem 4 (Triangle M to S)

• Scale factor S → M is \( \frac{3}{2} \), so M → S is the reciprocal: \( \frac{2}{3} \).

Problem 5 (Triangles A, B, C)

Compare sizes (A < B < C):

• a. A → B: B is larger, scale factor > 1.
• b. B → C: C is larger, scale factor > 1.
• c. B → A: A is smaller, scale factor < 1.
• d. C → A: A is smaller, scale factor < 1.

Final Answers:

1.

• Rectangle P to R: \( \boldsymbol{> 1} \)
• Rectangle Q to S: \( \boldsymbol{< 1} \)
• Rectangle Q to R: \( \boldsymbol{> 1} \)
• Rectangle S to P: \( \boldsymbol{> 1} \)
• Rectangle R to P: \( \boldsymbol{< 1} \)

1. Scale factor (S → L): \( \boldsymbol{2} \)

1. Scale factor (L → S): \( \boldsymbol{\frac{1}{2}} \)

1. Scale factor (M → S): \( \boldsymbol{\frac{2}{3}} \)

5.

• a. \( \boldsymbol{> 1} \)
• b. \( \boldsymbol{> 1} \)
• c. \( \boldsymbol{< 1} \)
• d. \( \boldsymbol{< 1} \)