Source: Official Guide for GMAT Review 2016 Data Sufficiency ; #126

1

In triangle ABC, point X is the

In triangle ABC, point X is the midpoint of side AC and point Y is the midpoint f side BC. If point R is the midpoint of line segment XC and if point S is the midpoint of line segment YC, what is the area of triangular region RCS?

2 Explanations

2

priyanshi jain

Does a median always divide the triangle into 2 equal parts (with equal area)?

Apr 21, 2016 • Comment

Cydney Seigerman, Magoosh Tutor

Hi Tarin :)

Did you mean to ask about the midpoint of a line segment? The median is the middle number in a given sequence of numbers. On the other hand, the midpoint is the middle point of a line. The midpoint divides a line segment into two line segments of equal length.

In this question, we're told that X is the midpoint of AC. This means that

AX = 1/2AC

With that in mind, we can find the area of the triangles ABC and ABX, given the fact that the height of both of these triangles is BX (the distance from the top vertex, B, to the base of the triangle). Using the formula for the area of a triangle, we have

area = 1/2*base*height
area ABC = 1/2*(AC)*(BX)
area ABX = 1/2*(AX)*(BX)

Replacing AX with 1/2AC, we have

area ABX = 1/2*(1/2*AC)*(BX) = 1/4*(AC)*(BX)

The ratio area ABC : area ABX is therefore

area ABC/area ABX
= [1/2*(AX)*(BX)]/[1/4*(AC)*(BX)]
= 2/1

We can see that we're able to relate the area of these two triangles. Thus, given the area of triangle ABX, we can determine the area of triangle ABC.

I hope this helps you clarify your doubts :)

Happy studying! :D

Apr 23, 2016 • Reply

PRANAVA SHEORAN

Hi Aman,
The median of a triangle always divides it into 2 triangles of equal area. (A median of a triangle is a line segment joining a vertex to the midpoint of the opposing side)

Nov 6, 2016 • Reply

Sam Kinsman

Hi Pranava,

You're right! In geometry, the "median" of a triangle is a line segment that joins vertex with the midpoint of the opposing side. The median goes through the center of the triangle, and divides the area of the triangle into two equal parts.

I've never seen the word "median" used this way on the GMAT - it's most often used to refer to the middle number in a given sequence of numbers. However, it's probably doesn't hurt to know the "geometry meaning" of median as well!

Nov 11, 2016 • Reply

3

Mike McGarry, Magoosh Tutor

Aug 18, 2015 • Comment

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