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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.

 

 Jun 18, 2024
 #1
avatar+1075 
+1

First, let's note that it is impossible for PQCD to be a square. 

We can use a simple calculation trick to find that \(QD = \sqrt{9(9 + 16)} = 15\).

Therefore, PQCD is a rectangle with sidelengths 16 and 15. 

 

Now, let's find the radius. 

We know the radius is half the diameter. With the dimensions given, we find that

the diameter, AB, is equal to \( 9 + 16 + 9 = 34\)

 

Thus, the radius is just half that, so we have 

\(\dfrac{34}2 = 17\)

 

So our final answer is 17. 

 

Thanks! :)

 Jun 18, 2024
 #2
avatar+1075 
+1

We have to find ABCD, which is a trapezoid. 

A trapezoid's area can be calculated with the equation \(\frac{(b1+b2)h}{2}\) where b1 and b2 are the bases and h is the height. 

We already have all the dimensions given. AB is b1, DC is b2, DQ is the height. 

So, we just plug in all the numbers and we should be set. Note that DQ is 15, as explained earlier. 

 

So, we have 

\(\frac{(DC+AB)DQ}{2} = (16+34)15/2\\ = \frac{50*15}{2} = 25*15=375\)

 

So our answer is just 375. 

 

Thanks! :)

 Jun 18, 2024

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