GEOMETRY - TRIANGLES

Question

This question is from an old GC thread.... Not sure what the right answer is

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 triangle region RCS
(1) the area of triangular region ABX is 32
(2) the length of one of the altitudes of triangle ABC is 8

scthakur · Answer

I will go with E since ABC can be an irregular triangle and the two statements do not give enough information.

KASSALMD · Answer

It should be A.
The solution is based on similarity of triangles and the ratio and proportion of the sides.
The area of the triangle RSC = 4

arjtryarjtry · Answer

the ans is E...
both do not satisfy

MRGiacalone · Answer

If you draw a line, in a triangle, going from the middle of two points, to the third point, you will always cut the triangle in half, no matter what type of triangle you have, since every new point we are creating in the triangle is half way between two points, we only have to divide the triangle until we get to the desired triangle, in this case RCS, which is the result of:

-Divide ABC in half, you will have two equal triangles ABX and CBX, we know ABX is equal to 32 so CBX should be 32 as well.
-Since RCS is in triangle CBX, lets divide CBX in two, easy Y is right in the middle of C and B, so a line from Y to X will result in triangles CYX and BYX, both equal with areas of 32/2=16 each
-Now we use triangle CYX, we have R right in the middle of C and X, draw a line from R to Y, both new triangles, RYX and CRY, will have an area of 16/2=8 each.
-By now the only logical thing to do is to divide CRY using midpoint of CY, called S, to draw a line to R. The two new triangles RSY and RSCeach with an area of 8/2=4

RSC=4, using statement (1), to me statement (2) is not sufficient, so my answer is A

amitdgr · Answer

Is this always true ?

Thanks

MRGiacalone · Answer

Yep, always. Ask your self this, what would be your approach to divide a triangle in two equal triangles?. Also you can draw a few triangles on a paper and use a ruler to confirm the theory.

Nerdboy · Answer

The way I drew it, triangles ABC, XYC and RSC are similar. Triangle ABX is not similar to them, and is not similar to XBC either. So answer should be E.

Nerdboy · Answer

Sorry, I just realised that ABX and CBX are not similar, but have the same area... so yes, it is A.

chayanika · Answer

How they have same area ?

BobbyRamos · Answer

i'm new here and i don't see the options. what is A? what is B?

triangle CXY is exactly 1/4 of the big triangle. note that if we let Z be the midpoint of side AB, then the four triangles AXZ, BYZ, CXY, and XYZ are all similar, congruent, and, therefore, with equal areas. similarly, if we assign T to be the midpoint of XY and we connect R, S, and T, we just have divided triangle CXY into four smaller, similar, and congruent triangles. area of triangle CRS is exactly 1/16 of the area of triangle ABC. the information in 2--an altitude of ABC--is not necessary.

vyassaptarashi · Answer

Clearly Option A is an ans

GEOMETRY - TRIANGLES

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