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