## In triangle ABC, point X is the midpoint…

## Solution:

We are given that in triangle ABC, point X is the midpoint of side AC, point Y is the midpoint

of side BC, point R is the midpoint of line segment XC and point S is the midpoint of line

segment YC. We need to determine the area of triangular region RCS. Let’s first sketch a

diagram based on the above information.

We’ve drawn line segment RS because we need to find the area of triangle RCS.

Even though we are not asked for the area of triangle XCY, we’ve drawn line

segment XY also because triangles ACB, XCY and RCS are all similar. In fact, the

base and height of each triangle is twice as long as the base and height of the next

triangle. (For example, AB is twice as long as XY, which is in turn twice as long as

RS.) Thus, the area of triangle ACB is 4 times the area of triangle XCY, which is in

turn 4 times the area of triangle RCS. That is, if we know the area of either triangle

ACB or triangle XCY, we will know the area of triangle RCS.

**Statement One Alone:**

The area of triangular region ABX is 32.

First let’s include line segment BX in the diagram (see above). We are given that

the area of triangle ABX is 32. Now let’s compare triangles ABX and XYB. These two

triangles have the same height (see dotted vertical lines below). However, we have

mentioned that AB (the base of triangle ABX) is twice as the length of XY (the base

of triangle XYB); therefore, the area of triangle ABX is also twice the area of

triangle XYB.

Since the area of triangle ABX is 32, the area of triangle XYB is 16. In other words,

the area of trapezoid ABXY is 32 + 16 = 48. Since the area of triangle ACB is 4

times the area of triangle XCY, if we let the area of XCY be x, then the area of

triangle ACB will be 4x. Also we can see that the area of triangle XCY plus the area

of trapezoid ABXY is the area of triangle ACB, so we have

x + 48 = 4x

48 = 3x

x = 16

Thus, we know that the area of triangle XCY is 16. Since we also know that the area

of triangle XCY is 4 times the area of triangle RCS, then the area of triangle RCS is

4. Statement one alone is sufficient to answer the question.

**Statement Two Alone:**

The length of one of the altitudes of triangle ABC is 8.

We are not given exactly which altitude of triangle ABC has length of 8 and, without

knowing the length of any bases of triangle ABC, we can’t know the area of triangle

ABC. Therefore, we can’t find the area of triangle RCS. Statement two alone is not

sufficient to answer the question.

**Answer: A**