Question

quadrilateral abcd is dilated to create quadrilateral abcd, as shown.
what is the scale factor of the dilation?

Explanation:

Step1: Recall scale factor formula

The scale factor of dilation is the ratio of the length of a side in the image (dilated figure) to the corresponding side in the original figure. So, \( \text{Scale Factor} = \frac{\text{Length in } A'B'C'D'}{\text{Length in } ABCD} \).

Step2: Choose a corresponding side pair

Let's take side \( AB \) (length 3.2) and its corresponding side \( A'B' \) (length 2.4). Calculate the ratio: \( \frac{2.4}{3.2} \).

Step3: Simplify the ratio

Simplify \( \frac{2.4}{3.2} \). Multiply numerator and denominator by 10 to eliminate decimals: \( \frac{24}{32} \). Then divide numerator and denominator by 8: \( \frac{24\div8}{32\div8} = \frac{3}{4} = 0.75 \)? Wait, no, wait. Wait, maybe I mixed up. Wait, dilation: if the image is smaller, scale factor is less than 1. Wait, let's check another pair. Let's take \( BC = 4.8 \) and \( B'C' = 3.6 \). Then \( \frac{3.6}{4.8} = \frac{36}{48} = \frac{3}{4} = 0.75 \)? Wait, no, wait 3.6 divided by 4.8: 3.6/4.8 = (36/10)/(48/10) = 36/48 = 3/4 = 0.75? Wait, but 2.4/3.2: 2.4 divided by 3.2. Let's do 2.4 ÷ 3.2. Multiply numerator and denominator by 10: 24 ÷ 32 = 0.75. Wait, or 4.2/5.6: 4.2 divided by 5.6. 4.2/5.6 = (42/10)/(56/10) = 42/56 = 3/4 = 0.75. And 2.7/3.6: 2.7/3.6 = (27/10)/(36/10) = 27/36 = 3/4 = 0.75. So the scale factor is 3/4 or 0.75? Wait, but the keypad has 1-9. Wait, maybe I made a mistake. Wait, 3.2 * 0.75 = 2.4, 4.8 * 0.75 = 3.6, 5.6 * 0.75 = 4.2, 3.6 * 0.75 = 2.7. Yes, that works. So the scale factor is 3/4, which is 0.75? Wait, but the keypad has 1-9. Wait, maybe I messed up the direction. Wait, dilation: if the original is ABCD and the image is A'B'C'D', then scale factor is image over original. So if A'B'C'D' is the image, then scale factor is (length in A'B'C'D') / (length in ABCD). So 2.4/3.2 = 0.75, which is 3/4. But 3/4 is 0.75, but the keypad has numbers 1-9. Wait, maybe I made a mistake. Wait, 3.2 * 0.75 = 2.4, correct. 4.8 * 0.75 = 3.6, correct. 5.6 * 0.75 = 4.2, correct. 3.6 * 0.75 = 2.7, correct. So the scale factor is 3/4, which is 0.75. But the keypad has 1-9. Wait, maybe the problem is written as a fraction? Wait, 3/4 is 0.75, but maybe the answer is 3/4, but the keypad has 3 and 4? Wait, no, the keypad is for input. Wait, maybe I miscalculated. Wait, 2.4 divided by 3.2: let's do this division. 2.4 ÷ 3.2 = (2.4 × 10) ÷ (3.2 × 10) = 24 ÷ 32 = 3 ÷ 4 = 0.75. So the scale factor is 0.75, which is 3/4. But the keypad has numbers 1-9. Wait, maybe the question is expecting 3/4, but in decimal, 0.75. But the keypad has 1-9, so maybe I made a mistake. Wait, wait, maybe I mixed up original and image. Wait, maybe the scale factor is original over image? No, dilation scale factor is image length over original length. Wait, let's check the definition: "The scale factor of a dilation is the ratio of the length of a side in the image to the length of the corresponding side in the original figure." So yes, image over original. So 2.4/3.2 = 0.75. So the scale factor is 0.75, which is 3/4. But the keypad has 1-9, so maybe the answer is 3/4, but in the keypad, we can input 0.75? Wait, but the keypad shows 1,2,3,4,5,6,7,8,9. Wait, maybe I made a mistake in the side lengths. Wait, AB is 3.2, A'B' is 2.4. 2.4/3.2 = 3/4. BC is 4.8, B'C' is 3.6. 3.6/4.8 = 3/4. CD is 3.6, C'D' is 2.7. 2.7/3.6 = 3/4. DA is 5.6, D'A' is 4.2. 4.2/5.6 = 3/4. So all ratios are 3/4. So the scale factor is 3/4, which is 0.75. But the keypad has numbers 1-9, so maybe the answer is 3/4, but in the box, we can write 0.75 or 3/4. Wait, but the problem is asking for the scale factor, so the answer is 3/4 or 0.75. But let's check the calculation again. 2.4 divided by 3.2: 2.4 ÷ 3.2 = 0.75. Yes. So the scale factor is 0.75, which is 3/4.

Answer:

\( \frac{3}{4} \) (or 0.75)