In triangle ABC, point X is the midpoint of side AC and : DS Archive
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# In triangle ABC, point X is the midpoint of side AC and

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In triangle ABC, point X is the midpoint of side AC and [#permalink]

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26 Apr 2007, 05:51
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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.
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26 Apr 2007, 10:18
Answer A, see attachement for explaination.
Attachments

SDOC3372.pdf [25.82 KiB]

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26 Apr 2007, 16:52
The OA is indeed A.

vijay2001, I really appreciate your help.

Your diagram and clear explanation makes it so easy to understand!

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26 Apr 2007, 18:04
I came up with A as well, but with a different value.

[1] Triangle ABX=32 --> bh/2=32 --> bh=64 and b=64/h

Therefore, segment XC = 64/h

Since we know that R is the midpoint of XC we can say that the base of triangle RCS is
RC=(64/h)(1/2) = 64/2h

Plug this into the equation for the area of a triangle (bh/2) and you get

(1/2)(64/2h)(h)=area of triangle RCS

Simplify and you 64h/4h, which = 16

[2] tells us that BX=8 or AY=8 or C?=8 insufficient

Since I got a different answer, anyone want to punch holes in my thought process?
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26 Apr 2007, 22:06
hi vijay, can you explain how area of triangle YRC = area of Z'RY ?

Thanks !
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27 Apr 2007, 19:36
Median of a triangle (line drawn from mid point of one side to opp angle) splits the triangle in to two equal areas. I have used this principle 3 times inorder to get area of triangle SRC.
27 Apr 2007, 19:36
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