Question:

In the figure, O is the center of the circle. Which one of the following must be true about the perimeter of the triangle shown?

(A) Always less than 10
(B) Always greater than 40
(C) Always greater than 30
(D) Always less than 30
(E) Less than 40 and greater than 20

“In the figure, O is the center of the circle. Which one of the following”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:
Approach Solution 1:
Perimeter of triangle OAB = OA + OB + AB = 10 + 10 + AB
= 20 + AB

3rd side property of a triangle:
3rd side is always greater than difference of the other two sides and less than sum of the other two sides
—> AB > 10 - 10
—> AB > 0
And
AB < 10 + 10
—> AB < 20

So, Perimeter is > 20 and < 40

Correct Answer: E

Approach Solution 2:
Given

• O is the centre of the circle as shown.

To find

• The option that is true about the perimeter of the triangle AOB

Approach and Working out

• AO = BO = Radius of the circle = 10
• Difference of other two sides < Third Side < Sum of other two sides
• 10 – 10 < Third side < 10 + 10
• 0 < Third side < 20
• 0 + length of other two sides < length of other two sides + third side < 20 + length of other two sides
• 20 < Perimeter < 40

Correct Answer: E

Approach Solution 3:
We can see that OA and OB, two of the triangle's sides, must each be 10 because they are also the circle's radii. Therefore, the third side, AB, needs to be more than 0 but less than 20. (the sum of the lengths of OA and OB). Therefore, the perimeter of the triangle must be greater than 10 + 10 + 0 = 20, but less than 10 + 10 + 20 = 40.

Correct Answer: E

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