Answer:
- ED = 2√19 inches
- A(EDCB) = 2√34 square inches
Step-by-step explanation:
The key to solving this problem is to recognize that the horizontal distance from AD to E is 2×AB, and the vertical distance from AB to E is 2×BC.* This gives rise to two equations using the Pythagorean theorem:
- AB² +BC² = 5²
- AB² +(2×BC)² = 7²
Subtracting the first equation from the second, we get ...
(AB² +4×BC²) -(AB² +BC²) = 49 -25
3×BC² = 24
BC = √(24/3) = √8
Then ...
AB² = 25 -8 = 17
AB = √17
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The length ED is also found using the Pythagorean theorem:
ED² = (2×AB)² +BC² = 4×17 + 8 = 76
ED = √76
ED = 2√19 . . . inches
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The area of triangle EDC is ...
A(EDC) = (1/2)AB×BC
and the area of triangle ECB is ...
A(ECB) = (1/2)BC×AB
So the area of EDCB is the sum ...
A(EDCB) = A(EDC) +A(ECB)
= (1/2)AB×BC +(1/2)AB×BC = AB×BC
= (√17)(√8)
A(EDCB) = 2√34 . . . square inches
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* You might be able to better see this if you translate rectangle ABCD so its diagonal AC is coincident with segment CE.
