Subject Test Math 1

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Subject Test Math 1

Question 31 of 32

In the figure, rectangle A B C D is inscribed in a circle; that is, the vertices A, B, C, and D lie on the circle. The rectangle A B C D partitions the circle into 5 regions, the rectangular region and four other non-overlapping regions outside the rectangle. The four non overlapping regions outside the rectangle and inside the circle are shaded.

Rectangle $A B C D$ is inscribed in the circle shown above. If the length of side $\bar{A B}$ is $5$ and the length of side $\bar{B C}$ is $12$ , what is the area of the shaded region?

Select an Answer

$40.8$

Correct Answer:

No

$53.1$

Correct Answer:

No

$72.7$

Correct Answer:

Yes

$78.5$

Correct Answer:

No

$81.7$

Correct Answer:

No

View Correct Answer

The correct answer is C.

The area of the shaded region can be found by subtracting the area of rectangle $A B C D$ from the area of the circle. To determine the area of the circle, first find the radius $r$ , and then compute the area $π r^{2}$ . Since rectangle $A B C D$ is inscribed in the circle, $∠ A B C$ is an inscribed right angle, and thus $\bar{A C}$ is a diameter of the circle. Applying the Pythagorean theorem to right triangle $A B C$ ,one finds the length of side $\bar{A C}$ is $\sqrt{5^{2} + 12^{2}} = \sqrt{169} = 13$ Thus the radius of the circle is $\frac{13}{2}$ , and the area of the circle is ${π (\frac{13}{2})}^{2} = \frac{169}{4} π$ . The area of rectangle $A B C D$ is $5 \times 12 = 60$ , and therefore, the area of the shaded region is $\frac{169}{4} π - 60 \approx 72.7$ .