fruti_yum wrote: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 ?
(1) The area of triangular region ABX is 32.
(2) The length of one of the altitudes of triangle ABC is 8.
Given: 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
Let's make a few
observations before we do anything else.
As others have pointed out before me, the triangles we get by connecting midpoints are
similar triangles.
Let's examine a specific case and then generalize...
Consider triangle ABC with these measurements...
The area of triangle ABC = (base)(height)/2 = (16)(12)/2 = 96
Now add in the midpoints X and Y...
This means triangle XCY is similar to triangle ACB
The area of triangle XYC = (base)(height)/2 = (8)(6)/2 = 24
In other words, the area of triangle XYC is 1/4 the area of triangle ABC
Now add midpoint R and S...
The area of triangle RCS = (base)(height)/2 = (4)(3)/2 = 6
In other words, the area of triangle RCS is 1/4 the area of triangle XYC
We can also say that the area of triangle RCS is 1/16 the area of triangle ABC
IMPORTANT: Since connecting midpoints (as we have done above) will always yield similar triangles, the results above (in green) will apply to all triangles.
Now onto the question....
Target question: What is the area of triangular region RCS ?
Statement 1: The area of triangular region ABX is 32
Our task is to determine whether there's a relationship between triangle ABX and triangle RCS
Let's see what triangle ABX looks like on our specific diagram....
If we let AX be the base, then we can see that the area of triangle ABX will be EQUAL to the area of triangle XCB, since both triangles have the same base and the same height.
Since triangles ABX and XCB have the same area, then we can also say that the area of triangle ABX is HALF the area of triangle ABC
So, if the area of triangle ABX is 32, then the area of triangular region ABC is 64
Since we already know that
the area of triangle RCS is 1/16 the area of triangle ABC, we can conclude that
the area of triangle RSC is 1/16 of 64, which equals 4
Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: The length of one of the altitudes of triangle ABC is 8
This just tells us one thing about triangle ABC.
Given this information, there's no way to determine the area of triangle ABC, which means there's no way to determine the area of triangle RCS
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Answer: A
Cheers,
Brent
