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

Question

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

KanishkM · Answer

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

Archit3110 · Answer

#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

EgmatQuantExpert · Answer

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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