Question

The parallelogram shown represents a map of the boundaries of a natural preserve. Walking trails run from points A to C and from points B to D. The measurements shown represent miles. Parallelogram A B C D is shown. Diagonals are drawn from point A to point C and from point D to point B and intersect at point E. The length of D E is y + 2, the length of E B is 3 y minus 4, and the length of E C is 2 y minus 3. What is the sum of the lengths of the two trails?

Answer 1

16 miles

Explanation:

Given parallelogram ABCD, you want the sum of the lengths of the two diagonals. Where E is their midpoint, you have BE=3y-4, CE=2y-3, DE=y+2.

Parallelogram

The diagonals of a parallelogram intersect at their midpoints. Hence opposite half-diagonals are congruent.

BE = DE

3y -4 = y +2

2y = 6 . . . . . . . . add 4-y

y = 3 . . . . . . . divide by 2

DE = y+2 = 3+2 = 5

BD = 2·DE = 2·5 = 10

CE

CE = 2(3) -3 = 3

AC = 2·CE = 2·3 = 6

Trails

The sum of the lengths of the trails is ...

AC +BD = 6 +10 = 16 . . . . miles

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