M08-22

Question

In triangle ABC, points M, N, and O are the midpoints of sides AB, BC, and AC, respectively. What is the area of triangle MON?

(1) The area of ABC is  \frac{\sqrt{3}}{4} 
 
(2) ABC is an equilateral triangle with a height of  \frac{\sqrt{3}}{2}

Bunuel · Accepted Answer

Official Solution:

In triangle ABC, points M, N, and O are the midpoints of sides AB, BC, and AC, respectively. What is the area of triangle MON? 
 
Look at the diagram below:
 
 https://gmatclub.com/gmat-practice-tests/resources/import/img/m08-22.png 
 
MN, NO, and OM are midsegments of triangle ABC (a midsegment is a line segment joining the midpoints of two sides of a triangle). An important property of a midsegment is:  the midsegment is always half the length of the third side.  So,  MN=\frac{AC}{2} ,  NO=\frac{AB}{2} , and  OM=\frac{BC}{2} .
 
Furthermore, since each side of triangle MNO is half of the side of triangle ABC, these triangles are similar (the ratio of all the sides is the same). An important property of similar triangles is:  if two similar triangles have sides in the ratio  \frac{x}{y} , their areas are in the ratio  \frac{x^2}{y^2} . 
 
Since the sides of the two similar triangles, MNO and ABC, are in the ratio 1:2, their areas are in the ratio 1:4. Therefore, (the area of MNO) = (the area of ABC)/4.
 
So, to find the area of MNO, we need to determine the area of ABC.
 
(1) The area of ABC is  \frac{\sqrt{3}}{4} . 
 
Sufficient.
 
(2) ABC is an equilateral triangle with a height of  \frac{\sqrt{3}}{2} .
 
We can determine the area of an equilateral triangle with the given height. Sufficient.

Answer: D

Senthil7 · Answer

I think this is a  high-quality  question and I agree with explanation.

Sanjay2043 · Answer

So how can we find the area of equilateral triangle by just knowing its height?

Bunuel · Answer

Check here: https://gmatclub.com/forum/everything-ab ... 87197.html

rahul2013 · Answer

I am bit confused with the Statement 1.

Since it is not given that the triangle is equilateral. I felt the statement 1 is insufficient,since any other type of triangle can also have the area given in stmt 1. 
Please help me out

Bunuel · Answer

The solution above shows that triangle MNO is similar to triangle ABC. Then it states a property which says that if two similar triangles have sides in the ratio  \frac{x}{y} , their areas are in the ratio  \frac{x^2}{y^2} . Therefore, since the sides of the two similar triangles, MNO and ABC, are in the ratio 1:2, their areas are in the ratio 1:4. So, the area of MNO equals one-fourth the area of ABC. Hence, to determine the area of MNO, we should find the area of ABC. (1) provides the area of ABC, so we can find the area of MNO.

rahul2013 · Answer

Thank you very much  Bunuel  . Its pretty clear now

leonardBe · Answer

I think this is a  high-quality  question and I agree with explanation.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

M08-22

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards