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In the given figure, a dart is thrown and lands in the interior of the circle. What is the probability that the dart will land in the shaded region?

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25 π − 48 25 π

25 π − 48 25 π

25 − 48 π 25 π

25 − 48 π 25 π

answer is A.

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Detailed Solution

Given, in the given figure, a dart is thrown and lands in the interior of the circle.
images (151×130)

Probability is defined as

f a v o u r a b l e area total area

.
Finding the hypotenuse,
We know,

A B = C D = 8

and

A D = B C = 6

Using pythagoras theorem in

△ ABC,

we get

A C 2 = A B 2 + B C 2

A C 2 = 82 + 62 = 100

A C = 10

O A = O C = 5

[since,

O

is the mid-point of AC.
Find the area of the shaded region,
[since,

O

is the mid-point of

AC]

Area of the circle

= π (OA) 2 = 25 π sq .

units
Area of rectangle

ABCD = AB × BC = 8 × 6 = 48 sq .

Units area of the shaded region =area of the circle - area of rectangle

ABCD = (25 π − 48)

sq. Units.
Now,

P (

dart lands in the shaded region

) = Area of shaded region Area of circle = 25 π − 48 25 π

.
The probability that the dart will land in the shaded region is

25 π − 48 25 π

.
Hence, the correct option is 1.

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