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Circle O with radius 2 is inscribed in a trapezoid ABCD such that AD is parallel to BC and CD = 8.  If angle A and angle B are right angles, then what is the area of ABCD?

 

 

 Dec 31, 2020
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Hello Guest!

 

A cool thing about circles inscribed in quadrilaterals are that the side thingys are equal. (Sorry for the bad description, here's a link) 

https://mathalino.com/reviewer/plane-geometry/quadrilateral-circumscribing-circle

 

Let's start by setting point X as where line OP intersects line AD and point Y where line OS intersects line DC. 

Let's also set the distance between XD as m and the distance YC as n. 

PD = DS = m

RC = SC = n

We also know that DS + SC = 8.

m + n = 8 

 

Because of the radius of the circle.

AP = AQ = QB = BR = 2 

 

To find the area of a trapezoid we need the height and the average of the two bases. 

The height of the trapezoid is AB = 2 + 2 (AQ + QB). 

The average of the bases in the trapezoid is (AP + PD + BR + RC)/2

We know AP = 2 and that BR = 2 too. 

PD + RC = m + n. 

m + n = 8

Plugging everything in, (2+2+8)/2 = 6. 

 

The average of the bases is 6 and the height is 4, making the area 24. 

I hope this helped. :))))

I'm not confident about my answer so it would be amazing if someone checked. 

 

=^._.^=

 Dec 31, 2020

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