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# help with trapezoid

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Trapezoid $ABCD$ is inscribed in the semicircle with diameter $\overline{AB}$, as shown below.  Find the radius of the semicircle.  Find the area of ABCD.

PQDC is a square.

Apr 30, 2024

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The diameter AB is 9 + 16 + 9 = 34. Therefore, the radius is half the diameter, which is $$\dfrac{34}2 = 17$$.

EDIT: There is one detail I missed, that is, it is not possible for PQDC to be a square. Simple calculations gives $$QD = \sqrt{9(9 + 16)} = 15$$, not 16. The quadrilateral PQDC is a rectangle with side lengths 16 and 15.

Apr 30, 2024
edited by MaxWong  Apr 30, 2024
#2
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For part b,

Like you already said, the dimater of the halfcircle is 34. We already know the height of the trapezoid since PQDC is a square. We also know DC is 16.

Now, we have everything needed for the area of the trapezoid, so let's begin!

We know the area of the trapezoid is $$\frac{(b1+b2)h}{2}$$ where b1 and b2 are the bases and the h is the height.

We have $$\frac{(25+16)16}{2} = (31)8 = 248.$$

So the area of the trapezoid is 248!

Thanks!

EDIT: Yes, Max Wong is right about the fact that PQDC cannot be a square. Plugging the new numbers in gets us an area of 232.5

Apr 30, 2024
edited by NotThatSmart  Apr 30, 2024