Given trapezoid ABCD with the measure of \(\angle{D}\) (in degrees) equals 45, *AD = \(8\sqrt{2}\), AB* = 4**, **BC = 10. Find the area of the trapezoid.

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I used trigonometry, and I got the answer of 88 (please check). However, this problem shouldn't use trig and shouldn't use the calculator. Can someone tell me how to solve it?

CalculatorUser Jul 13, 2019

#1**+5 **

Draw a line segment from A perpendicular to DC that meets DC at point E , and

draw a line segment from B perpendicular to DC that meets DC at point F , like this:

The sum of the interior angles in △ADE = 180°

45° + 90° + m∠DAE = 180°

m∠DAE = 180° - 45° - 90°

m∠DAE = 45°

△ADE is an isosceles triangle and the sides opposite the base angles are congruent.

DE = AE

And if DE = b then AE = b

By the Pythagorean Theorem,

DE^{2} + AE^{2} = (8√2)^{2}

b^{2} + b^{2} = (8√2)^{2}

2b^{2} = 128

b^{2} = 64

b = 8

ABFE is a rectangle so

BF = AE = 8

EF = AB = 4

By the Pythagorean Theorem,

BF^{2} + FC^{2} = 10^{2}

8^{2} + FC^{2} = 10^{2}

FC^{2} = 10^{2} - 8^{2}

FC^{2} = 36

FC = 6

Now we know:

AB = 4

So we can say base_{1} = 4

DE = 8 and EF = 4 and FC = 6

So we can say base_{2} = 8 + 4 + 6 = 18

AE = 8

So we can say height = 8

area of trapezoid = (1/2)(base_{1} + base_{2})(height)

area of trapezoid = (1/2)(4 + 18)(8)

area of trapezoid = 88

If you have the 45-45-90 triangle memorized, you can immediately recognize that DE and AE must be 8,

but if not you can always figure it out this way.

hectictar Jul 13, 2019

#2**+3 **

Thank you! This should've been easy, I will have to remember about isosceles triangles and Pythagorean theorem.

CalculatorUser
Jul 15, 2019