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Trapezoid $ABCD$ is inscribed in the semicircle with diameter $\overline{AB}$, as shown below.  If $CD = 7$, $AD = 5$, and $BC = 2$, then find the radius of the semicircle.

 Dec 17, 2023
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To solve this problem, we can follow these steps:

 

1. Identify relevant relationships:

 

Since ABCD is inscribed in the semicircle, radius r is half the diameter AB, i.e., r = AB/2.

We can use the Pythagorean theorem in triangles ADC and CBD to relate side lengths.

 

2. Apply the Pythagorean theorem in triangle ADC:

 

We know AD = 5 and CD = 7. Let AC = x. Applying the Pythagorean theorem:

 

x^2 + 5^2 = 7^2

x^2 = 24

x = 2sqrt(6)

 

3. Apply the Pythagorean theorem in triangle CBD:

 

We know BC = 2 and CD = 7. Let BD = y. Applying the Pythagorean theorem:

y^2 + 2^2 = 7^2

y^2 = 45

y = 3sqrt(5)

 

4. Find AB using side lengths from triangles:

 

AB = AD + BD = 5 + 3sqrt(5)

 

5. Calculate the radius:

 

r = AB/2 = (5 + 3sqrt(5))/2

 

Therefore, the radius of the semicircle is (5 + 3sqrt(5))/2.

 Dec 17, 2023

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