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What is the area of the circle with center M shown above? (1) The leng

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What is the area of the circle with center M shown above? (1) The leng  [#permalink]

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New post 18 Feb 2019, 09:50
2
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A
B
C
D
E

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What is the area of the circle with center M shown above?

(1) The length of AC is \(8 \sqrt{2}\)
(2) The length of arc ABC is \(4π\)

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Re: What is the area of the circle with center M shown above? (1) The leng  [#permalink]

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New post 18 Feb 2019, 19:42
1

Solution



Given:
    • A circle with the center at point M, as shown in the diagram

To find:
    • The area of the circle

Approach and Working:
To determine the area of the circle, we need to know the length of the radius of the circle.

Analysing Statement 1
As per the information given in statement 1, the length of AC is 8√2.

In the triangle MAC,
    • MA = MC = radius of the circle = r
    • And, angle AMC = 90°

Hence, triangle MAC is an isosceles right-angled triangle.

So, applying Pythagoras Theorem, we can write
    • \({MA}^2 + {MC}^2 = {AC}^2\)
    Or, \(r^2 + r^2 = (8√2)^2\)

From this equation, we can find the value of r.

Hence, statement 1 is sufficient to answer the question.

Analysing Statement 2
As per the information given in statement 2, the length of arc ABC is 4π

As arc ABC creates 90° at the center, we can write it as
    • \(\frac{90}{360}\) x 2πr = 4π

From this equation, we can find the value of r.

Hence, statement 2 is sufficient to answer the question.

Hence, the correct answer is option D.

Answer: D

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Re: What is the area of the circle with center M shown above? (1) The leng  [#permalink]

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New post 18 Feb 2019, 10:11
SajjadAhmad wrote:
Attachment:
123.jpg


What is the area of the circle with center M shown above?

(1) The length of AC is \(8 \sqrt{2}\)
(2) The length of arc ABC is \(4π\)


IMO D

from 1) length of AC is given, now from this we can calculate the radius -> area
Since AM = AC, it becomes an isosceles triangle

from 2) length of arc is given
from here again we can calculate the radius -> area

theta/360 * 2\(π\) r, can be used
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Re: What is the area of the circle with center M shown above? (1) The leng  [#permalink]

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New post 18 Feb 2019, 10:37
SajjadAhmad wrote:
Attachment:
123.jpg


What is the area of the circle with center M shown above?

(1) The length of AC is \(8 \sqrt{2}\)
(2) The length of arc ABC is \(4π\)



#1
its an isoscles triangle
x^2 +x^2 = (8√2)^2
x^2 = 64
x= 8 = radius
sufficient
#2
90/360 * 2 * pi * r = 4pi
solve for r
r = 8
sufficient
IMO D
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Re: What is the area of the circle with center M shown above? (1) The leng   [#permalink] 18 Feb 2019, 10:37
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