Question

What is the perimeter of angle DEF to the nearest tenth of a unit?
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Answer 1

Answer: 36.4 units (choice A)

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Step-by-step explanation:

Let's use the distance formula to find the distance from D to E

`D = (x_1,y_1) = (-7,2) \text{ and } E = (x_2, y_2) = (3,4)\\\\d = √((x_1 - x_2)^2 + (y_1 - y_2)^2)\\\\d = √((-7-3)^2 + (2-4)^2)\\\\d = √((-10)^2 + (-2)^2)\\\\d = √(100 + 4)\\\\d = √(104)\\\\d = √(4*26)\\\\d = √(4)*√(26)\\\\d = 2√(26)\\\\d \approx 10.198\\\\`

Note: uppercase D refers to the point, while lowercase d is the distance from D to E.

The length of segment DE is roughly 10.198 units long.

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Repeat for the distance from E to F.

`E = (x_1,y_1) = (3,4) \text{ and } F = (x_2, y_2) = (5,-7)\\\\d = √((x_1 - x_2)^2 + (y_1 - y_2)^2)\\\\d = √((3-5)^2 + (4-(-7))^2)\\\\d = √((3-5)^2 + (4+7)^2)\\\\d = √((-2)^2 + (11)^2)\\\\d = √(4 + 121)\\\\d = √(125)\\\\d = √(25*5)\\\\d = √(25)*√(5)\\\\d = 5√(5)\\\\d \approx 11.1803\\\\`

Segment EF is roughly 11.1803 units long.

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Repeat for the distance from F to D.

`F = (x_1,y_1) = (5,-7) \text{ and } D = (x_2, y_2) = (-7,2)\\\\d = √((x_1 - x_2)^2 + (y_1 - y_2)^2)\\\\d = √((5-(-7))^2 + (-7-2)^2)\\\\d = √((5+7)^2 + (-7-2)^2)\\\\d = √((12)^2 + (-9)^2)\\\\d = √(144 + 81)\\\\d = √(225)\\\\d = 15\\\\`

Unlike the others, this result is exact.

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Add up the three segment lengths to get the perimeter

DE + EF + FD

10.198 + 11.1803 + 15

36.3783

The perimeter is approximately 36.3783 units which rounds to 36.4

The answer has been confirmed with GeoGebra.