In Triangle ABC, Point X is the Midpoint of Side AC and Point Y is the Midpoint of Side BC GMAT Data Sufficiency

Question: In triangle ABC, point X is the midpoint of side AC and point Y is the midpoint of 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 ?

The area of triangular region ABX is 32.
The length of one of the altitudes of triangle ABC is 8.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: A

Solution and Explanation:
Approach Solution 1:
Let us take a look at the diagram below.

We will look at the midsegments of the triangles here. Midsegment is a line joining the midpoints of two sides of a triangle.

We see that XY is the midsegment of the bigger triangle ABC.
RS is the midsegment of the smaller triangle XYC.

Several important properties:

The midsegment is always half the length of the third side.
Therefore, AB/XY = 2 and XY/RS = 2
=> AB/RS = 4
As we know that the midsegment always divides a triangle into two similar triangles. So, ABC is similar to XYC and XYC is similar to RSC
Since, ABC - XYC - RSC, we can conclude that,
--> ABC is similar to RSC
According to above equation 1, the ratio of their sides is 4:1
If two similar triangles have sides in the ratio x/y, then their areas are in the ratio X^2/Y^2.
So, from above, we already know that ABC is similar to RSC and the ratio of their sides is 4:1.
The equation stands as: area ABC/ area RSC =\(4^2\) =16
So the area of ABC is 16 times as large as the area of RSC;
Each median divides the triangle into two smaller triangles which have the same area. Therefore, since X is the midpoint of AC then BX is the median of ABC. The area of ABX is therefore half of the area of ABC.
From the above, we have that the area of ABX is 16/2 = 8 times as large as the area of RSC.

So, to find the area of RSC we need to find the area of ABX.

Statement 1 alone:
The area of triangular region ABX is 32
--> the area of RSC=32/8=4. Therefore, Sufficient.
Statement 2 alone:
The length of one of the altitudes of triangle ABC is 8.
We cannot get the area with this sufficient information. Therefore, Not sufficient.

Approach Solution 2:
The problem statement states that:
Given:

In triangle ABC, point X is the midpoint of side AC and point Y is the midpoint of side BC.
Point R is the midpoint of line segment XC
Point S is the midpoint of line segment YC.

Find out :

the area of triangular region RCS.

This question can be solved by using the Midpoint theorem in the case of triangles.

As per the theorem, the segment joining the midpoints of two sides of a triangle is half the length of the third side. The smaller triangle thus formed is similar to the original triangle.

The ratio of sides of the smaller △ to the larger △= 1:2

=> Therefore, Area of smaller △ : Area of Larger △ = 1:4
From the given information we get:
Area(△CYX) : Area(△ABC) = 1:4
Area(△CSR) : Area(△CYX) = 1:4
=> Area(△CSR) = 1/16 * Area(△ABC)
Statement 1: The area of triangular region ABX is 32.
Area(△ABX) = 1/2 * Area(△ABC) - Since they have identical height and the base of ABX is half the base of ABC.
Thus from Area (△ABX), we can calculate Area (△CSR) => Area(△ABX)/8 = 4
=> SUFFICIENT
Statement 2: The length of one of the altitudes of triangle ABC is 8.
We cannot infer anything from the length of one of the heights.
=> NOT SUFFICIENT

Approach Solution 3:
The problem statement informs that:
Given:

In triangle ABC, point X is the midpoint of side AC and point Y is the midpoint of side BC.
Point R is the midpoint of line segment XC
Point S is the midpoint of line segment YC.

The question asked to find the area of the triangular region RCS.

The problem statement is testing the properties of similar triangles here. There are 3 ways to infer if the triangles are identical:
1. AAA (angle angle angle)
All three pairs of corresponding angles are identical.
(We actually require two angles)

2. SSS in the same proportion (side side side)
All three pairs of corresponding sides are in identical proportion

3. SAS (side angle side)
Two pairs of sides in identical proportion and the included angle equal.

The 3rd property is quite suitable here.

Two triangles ABC and RSC are identical since RC/AC = SC/BC = ¼. They include the same angle THETA.
1) The area of triangular region ABX is 32.
This implies that Area(△ABX) = 32.

Since the height of △ ABC is identical to that of △ABX and the base is twice, we get:
the area (△ABC) = 32*2= 64.
Since △ABC and △RSC are identical and their sides are in the ratio of 1:4, their areas will be in the ratio of 1:16. – Sufficient.
2) The length of one of the altitudes of triangle ABC is 8.
This implies one of the altitudes is 8. Since it does not tell which one, the statement is insufficient.

“In triangle ABC, point X is the midpoint of side AC and point Y is the midpoint of side BC”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide 2021". The GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions come up with a problem statement that is followed by two factual statements. GMAT data sufficiency includes 15 questions which are two-fifths of the entire 31 GMAT quant questions.

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In Triangle ABC, Point X is the Midpoint of Side AC and Point Y is the Midpoint of Side BC GMAT Data Sufficiency

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