Question

1. find the scale factor of figure a to figure b

Explanation:

Step1: Identify corresponding sides

Find two corresponding sides of Figure A and Figure B. For example, side of A: 36, side of B: 20? Wait, no, wait. Wait, Figure A has sides 36 and 25.2? Wait, no, Figure B has sides 14 and 20? Wait, maybe I misread. Wait, Figure A: let's check the sides. Wait, the problem is to find scale factor from A to B. So scale factor is (length of B)/(length of A) for corresponding sides. Let's find corresponding sides. Let's see, maybe 25.2 in A corresponds to 14 in B? Wait, 25.2 and 14. Let's check 25.2 ÷ 14 = 1.8? Wait, no, scale factor from A to B is (B's side)/(A's side). Wait, maybe another pair. Wait, 36 and 20? No, 36 ÷ 20 = 1.8, but 25.2 ÷ 14 = 1.8. Oh, right! So 25.2 (A) and 14 (B)? Wait, no, scale factor from A to B is (length of B)/(length of A). Wait, no: scale factor from A to B is (B's length)/(A's length) if A is the original and B is the image. Wait, let's confirm: scale factor = (length of image)/(length of original). So if we go from A to B, B is the image, A is original. So find corresponding sides. Let's take the side of A as 25.2 and side of B as 14? Wait, 25.2 ÷ 14 = 1.8? No, wait, 14 ÷ 25.2? Wait, no, maybe I got the sides reversed. Wait, Figure A: let's look at the labels. Figure A has a side labeled 36 and 25.2? Wait, Figure B has 14 and 20. Wait, maybe 36 (A) and 20 (B)? No, 36 ÷ 20 = 1.8, 25.2 ÷ 14 = 1.8. Ah, so 25.2 in A corresponds to 14 in B? Wait, no, 25.2 is A, 14 is B. So scale factor from A to B is 14/25.2? Wait, no, that would be less than 1. Wait, maybe I mixed up. Wait, maybe 25.2 is B and 14 is A? No, the figure: Figure A is larger, Figure B is smaller. So scale factor from A to B is (B's length)/(A's length). So let's take corresponding sides. Let's take the side of A as 25.2 and side of B as 14. Then scale factor is 14 / 25.2. Simplify: 14 ÷ 25.2 = 140 ÷ 252 = divide numerator and denominator by 28: 5 ÷ 9? No, wait 14 and 25.2: 25.2 = 14 × 1.8, so 14 = 25.2 ÷ 1.8. Wait, 25.2 ÷ 14 = 1.8. So if A is the original, B is the image, then scale factor is 14/25.2? Wait, no, that can't be. Wait, maybe I have the sides reversed. Wait, maybe 36 (A) and 20 (B)? 36 ÷ 20 = 1.8, 25.2 ÷ 14 = 1.8. So 36 (A) and 20 (B)? No, 20 is B, 36 is A. So 20/36 = 5/9? No, that's not 1.8. Wait, no, 25.2 and 14: 25.2 ÷ 14 = 1.8, so 14 × 1.8 = 25.2. So that means A is 1.8 times B? No, wait, no: if A has a side of 25.2 and B has 14, then to get from A to B, we divide by 1.8, so scale factor is 14/25.2 = 5/9? Wait, no, 14 ÷ 25.2 = 140 ÷ 252 = 5 ÷ 9 ≈ 0.555... But 25.2 ÷ 14 = 1.8. So maybe the corresponding sides are 36 (A) and 20 (B)? 36 ÷ 20 = 1.8, 25.2 ÷ 14 = 1.8. So that means 36 (A) corresponds to 20 (B)? No, 36 and 20: 36 is A, 20 is B. So 20/36 = 5/9 ≈ 0.555, but 25.2/14 = 1.8. Wait, I think I messed up the correspondence. Let's look at the triangles. They are similar, so corresponding sides are proportional. Let's list the sides:

Figure A: let's say the two sides are 36 and 25.2.

Figure B: the two sides are 20 and 14.

Wait, 36/20 = 1.8, 25.2/14 = 1.8. Ah! So 36 (A) corresponds to 20 (B)? No, 36 is A, 20 is B? Wait, no, 36 ÷ 20 = 1.8, 25.2 ÷ 14 = 1.8. So that means the ratio of A to B is 1.8, but scale factor from A to B is (B's length)/(A's length) = 20/36 = 5/9? Wait, no, that's not. Wait, scale factor is defined as (length of image)/(length of original). So if we are going from A to B, B is the image, so scale factor is (length of B)/(length of A) for corresponding sides. So if 36 is A's side, 20 is B's side, then scale factor is 20/36 = 5/9 ≈ 0.555... But 25.2/14 = 1.8, which is the reciprocal. So I must have the correspondence wrong. Wait, maybe 25.2 is B's side and 14 is A's? No, the figure: Figure A is larger, so A has longer sides. So 25.2 is A, 14 is B. So scale factor from A to B is 14/25.2. Let's compute that: 14 ÷ 25.2. Multiply numerator and denominator by 10: 140 ÷ 252. Divide numerator and denominator by 28: 5 ÷ 9 ≈ 0.555... But 25.2 ÷ 14 = 1.8, which is 9/5. So that would be scale factor from B to A. So the problem is "Find the scale factor of Figure A to Figure B", so A to B: (B's length)/(A's length). So let's take the corresponding sides: A has 25.2, B has 14. So 14/25.2 = 5/9 ≈ 0.555... Wait, but 36 and 20: 20/36 = 5/9. So that's consistent. So 36 (A) and 20 (B): 20/36 = 5/9. 25.2 (A) and 14 (B): 14/25.2 = 5/9. Yes, because 25.2 × (5/9) = 25.2 ÷ 9 × 5 = 2.8 × 5 = 14. Correct. And 36 × (5/9) = 20. Correct. So the scale factor from A to B is 5/9 or approximately 0.555... Wait, but let's check the calculation again. 14 divided by 25.2: 25.2 ÷ 14 = 1.8, so 14 = 25.2 ÷ 1.8, so 14/25.2 = 1/1.8 = 5/9. Yes, because 1.8 = 9/5, so reciprocal is 5/9. So scale factor from A to B is 5/9.

Step2: Calculate the scale factor

Take the length of a side in Figure B and divide by the corresponding length in Figure A. Let's use the sides 14 (B) and 25.2 (A). So scale factor = 14 / 25.2. Simplify:

14 / 25.2 = (14 × 10) / (25.2 × 10) = 140 / 252

Divide numerator and denominator by 28: 140 ÷ 28 = 5, 252 ÷ 28 = 9. So 5/9.

Alternatively, use 20 (B) and 36 (A): 20 / 36 = 5/9. Same result.

Answer:

$\frac{5}{9}$ (or approximately 0.555...)