Answer:
- CD = 1.5 miles
- AE = 12.5 miles
Explanation:
Given the figure with triangle ABC similar to triangle EDC and AB=8 mi, ED=2 mi, BD = 7.5 mi, you want the measures of CD and AE.
b. CD
Similar triangles will have corresponding sides proportional. That means ...
ED/CD = AB/CB
2/CD = 8/(7.5 -CD)
Inverting the ratios and multiplying by 8 gives ...
4·CD = 7.5 -CD
5·CD = 7.5 . . . . . . . add CD
CD = 1.5 . . . . . . . . . divide by 5
c. AE
The distance AE is the hypotenuse of a right triangle with side lengths 7.5 and (8+2) = 10. The Pythagorean theorem can be used to find AE:
AE² = 7.5² +10² = 56.25 +100 = 156.25
AE = √156.25 = 12.5
AE = 12.5 miles, the distance to the mall.
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Additional comment
You may recognize these triangles are 3-4-5 triangles. ABC has a scale factor of 2, so has side lengths 6-8-10. EDC has a scale factor of 1/2, so has side lengths 1.5, 2, 2.5. The triangle with AE as its hypotenuse is the sum of these, so has a scale factor of 2.5 (miles).
AE = (2.5 miles) · 5 = 12.5 miles