Question

figure b is a scaled copy of figure a. for each set of figures, calculate the scale factor: • from figure a to figure b. • from figure b to figure a. 1. image of a and b on grid 2. image of a and b on grid 3. image of a and b on grid

Explanation:

Problem 1

Step1: Determine side lengths of A and B

Assume each grid square has side length 1. For figure A (the square part), let's say a side is 3 units. For figure B, the corresponding side is 1 unit (since it's smaller).

Step2: Scale factor A to B

Scale factor from A to B is $\frac{\text{Length of B}}{\text{Length of A}} = \frac{1}{3}$? Wait, no, wait. Wait, maybe I got A and B reversed. Wait, looking at the first figure: Figure A has a square with side 3 (assuming grid), and Figure B has a square with side 1? Wait, no, maybe the vertical line: Figure A's vertical line is, say, 6 units, and Figure B's is 2? Wait, maybe better to count grid squares. Let's take the square part: Figure A's square is 3x3 (3 units per side), Figure B's square is 1x1? No, wait, the first figure: Figure A has a square with side length 3 (from grid), Figure B has a square with side length 1? Wait, no, maybe the scale is 3:1? Wait, no, let's re-examine. Wait, the problem says Figure B is a scaled copy of A. So if A is larger, then scale factor from A to B is less than 1, and from B to A is greater than 1. Let's take the square in A: let's say it's 3 units (3 grid squares), and in B it's 1 unit (1 grid square). So:

Step1: Find corresponding side lengths

Let side of A's square = 3, side of B's square = 1.

Step2: Scale factor A to B

Scale factor = $\frac{\text{Length of B}}{\text{Length of A}} = \frac{1}{3}$? Wait, no, wait: scale factor from A to B is (length of B)/(length of A). Wait, if A is the original, B is the copy. So if A's side is 3, B's is 1, then scale factor A to B is 1/3. Then from B to A, it's 3/1 = 3.

Wait, maybe I made a mistake. Let's check again. Let's take the vertical line: in Figure A, the vertical line is, say, 6 units (6 grid squares), and in Figure B, it's 2 units. Then 2/6 = 1/3. So yes, scale factor A to B is 1/3, B to A is 3.

Step1: Determine corresponding lengths

Let's find a corresponding side. In Figure A, a certain segment is, say, 2 units, and in Figure B, it's 4 units? Wait, no, Figure B is larger. Let's count grid squares. Let's take the horizontal segment in A: suppose it's 2 units, and in B it's 4 units? Wait, no, let's look at the shape. Figure A is a smaller version, Figure B is larger. Let's take a side: in A, a side is 2 grid squares, in B, it's 4? Wait, no, maybe 1 to 3? Wait, no, let's do it properly. Let's take the horizontal length of the rectangular part in A: let's say it's 2 units, and in B it's 6 units? Wait, no, maybe the scale factor is 3? Wait, no, let's count. Let's take the vertical segment in A: 2 units, in B: 6 units? No, maybe 1 to 3. Wait, let's see: Figure A has a shape with a horizontal length of 2 (grid squares), Figure B has 6. So 6/2 = 3. Wait, no, scale factor from A to B is (length of B)/(length of A). So if A's length is 2, B's is 6, then scale factor A to B is 6/2 = 3? Wait, no, that would mean B is larger. Wait, the problem says Figure B is a scaled copy of A, so if B is larger, then scale factor A to B is greater than 1. Wait, maybe I had it reversed earlier. Wait, in problem 1, maybe A is smaller? No, the first figure: Figure A is on the left, larger, Figure B on the right, smaller. In problem 2, Figure A is on the left, smaller, Figure B on the right, larger. So let's correct.

Step1: Find corresponding side lengths

In Figure 2: Let's take a horizontal segment in A: let's say it's 2 grid units, and in B it's 6 grid units. Or a vertical segment: A has 2, B has 6. So length of B is 3 times length of A.

Step2: Scale factor A to B

Scale factor = $\frac{\text{Length of B}}{\text{Length of A}} = \frac{6}{2} = 3$? Wait, no, wait: if A is the original, B is the copy. So if A's length is 2, B's is 6, then scale factor A to B is 6/2 = 3. Then from B to A, it's 2/6 = 1/3.

Wait, let's confirm with another segment. In A, the vertical part: let's say it's 3 units, in B it's 9 units. 9/3 = 3. Yes, so scale factor A to B is 3, B to A is 1/3.

Step1: Determine side lengths

Let's take the length of Figure A (rectangle): suppose it's 9 grid units, and Figure B (rectangle) is 3 grid units. Or width: A's width is 4, B's is 4/3? No, wait, let's count. Let's say Figure A has length 9 (horizontal) and Figure B has length 3. So:

Step2: Scale factor A to B

Scale factor = $\frac{\text{Length of B}}{\text{Length of A}} = \frac{3}{9} = \frac{1}{3}$? Wait, no, wait: Figure A is larger, Figure B is smaller. So length of A: let's say 9, length of B: 3. So scale factor from A to B is 3/9 = 1/3. Then from B to A, it's 9/3 = 3.

Wait, let's check the width. If A's width is 4, B's width is 4/3? No, maybe the length is 10 and 2? Wait, no, let's look at the grid. Let's assume each grid is 1 unit. Figure A: length (horizontal) is 10 units, Figure B: length is 2 units? No, maybe 9 and 3. So 3/9 = 1/3. So:

Step1: Find corresponding lengths

Let length of A = 9, length of B = 3.

Step2: Scale factor A to B

$\frac{3}{9} = \frac{1}{3}$

Step3: Scale factor B to A

$\frac{9}{3} = 3$

Answer:

From A to B: $\frac{1}{3}$; From B to A: $3$

Problem 2