Question

△ade is a dilation of △abc with scale factor 1.25 and center of dilation at a. which statement is true? image of triangle with labels a, b, d on the base (ab=8, bd=2), ac=6, ce=5, and e at the top options: ad = (5/4)bd; (ab/bd) = (ae/ce); (ab/bc) = (ad/de); ab = (5/4)ac

Explanation:

Step1: Calculate AD

AD = AB + BD = 8 + 2 = 10

Step2: Check AD = 5/4 BD

BD = 2, 5/4 BD = 5/4 * 2 = 2.5 ≠ 10 → False

Step3: Check dilation ratio

Scale factor k = 1.25 = 5/4. For dilation center A, AC/AE = AB/AD = 1/k? No: AC=6, AE=AC+CE=6+5=11≠6(5/4)=7.5. Wait: Dilation maps ABC→ADE, so AD/AB = AE/AC = k. AD=10, AB=8 → 10/8=5/4=k. AE should be ACk=6(5/4)=7.5, but CE=AE-AC=7.5-6=1.5≠5? No, wait: Dilation preserves ratios of corresponding sides. So AB/AD = BC/DE = 1/k? AB=8, AD=10 → 8/10=4/5. BC/DE should be 4/5. Is there another way? Wait option C: AB/BC = AD/DE? If dilation, AB/AD = BC/DE → cross multiply: ABDE=AD*BC → AB/BC=AD/DE. Yes, that's equivalent.

Answer:

C. $\frac{AB}{BC} = \frac{AD}{DE}$