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ABCD is a rectangle inscribed in a circle. What is the area of the shaded region in the above figure if AB = 8cm and BC = 6cm?

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29 c m^{2}

30.5 c m^{2}

25 c m^{2}

32.5 c m^{2}

answer is B.

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Detailed Solution

Given, ABCD is a rectangle inscribed in a circle and

AB = 8

cm and

BC = 6

cm.

Pythagoras theorem states that the square of the hypotenuse side in a right-angled triangle equals the sum of the squares of the other two sides.
In

△ ABC

, apply pythagoras theorem.
Thus,

(AC)^{2} = (AB)^{2} + (AC)^{2}

\Rightarrow

(AC)^{2}

= 8^{2} + 6^{2}

\Rightarrow

(AC)^{2}

= 100

\Rightarrow

AC = 10

.
AC is the diameter of the circle.
Thus, radius=

\frac{diameter}{2}

\Rightarrow

Radius

= \frac{AC}{2}

\Rightarrow

Radius

= \frac{10}{2}

\Rightarrow

Radius

= 5

cm
Area of a rectangle is length×breadth.

\Rightarrow

Area of the rectangle ABCD

= 8 \times 6

\Rightarrow

Area of the rectangle ABCD

= {48 cm}^{2}

Area of the circle=

π r^{2}

\Rightarrow

Area of the circle

= π (5)^{2}

\Rightarrow

Area of the circle

= 3.14 \times 25

\Rightarrow

Area of the circle

= {78.5 cm}^{2}

Thus, the area of the shaded portion = Area of the circle - Area of the rectangle ABCD.

\Rightarrow

The area of the shaded portion =

(78.5 - 48) c m^{2}

\Rightarrow

The area of the shaded portion =

{30.5 cm}^{2}

Hence, option 2 is correct.

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ABCD is a rectangle inscribed in a circle. What is the area of the shaded region in the above figure if AB = 8cm and BC = 6cm?

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