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# I am not sure if someone has done this problem on this

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I am not sure if someone has done this problem on this [#permalink]  19 Jan 2010, 22:04
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Question Stats:

50% (01:25) correct 50% (03:30) wrong based on 6 sessions
I am not sure if someone has done this problem on this forum, but here it is:

109. 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.

Can someone please provide a picture of the solution? The OG shows it one way, but I feel like I would never have came to this solution using their method. Does someone have an easier method?
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Re: OG #12 Problem 109 [#permalink]  19 Jan 2010, 22:27
IMO ans should be A.

Area of triangle ABX = area of BXC as median divides the triangle into 2 equal areas.

similarly area of XYC = 1/2 BXC = ( 1/2 ABX) = 1/2 * 32 = 16

Now XYC and RSC are similar triangles
Since we have area of XYC , we can find the area of triangle RSC.
Thus statement 1 is sufficient.

Statement 2 seems not sufficient, not enough parameters given.

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Re: OG #12 Problem 109 [#permalink]  19 Jan 2010, 22:57
You are correct. The official answer is A.
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Re: OG #12 Problem 109 [#permalink]  19 Jan 2010, 23:05
Okay, I didn't see that. So, if you draw a line from the vertex of a triangle to the midpoint of an opposite line, does that automatically split the triangle into two smaller equal triangles? If so, the problem makes a lot more sense. Thanks for the explanation.
CEO
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Re: OG #12 Problem 109 [#permalink]  19 Jan 2010, 23:13
Yes the line joining mid point and opp vertex divides it into 2 triangles having equal area.

Area of triangle = 1/2 base * height

since height remains the same and base is divided into 2 equal parts...thus areas are also divided into 2 equal parts..
Letme know if u got the concept else I will elaborate it.
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Re: OG #12 Problem 109 [#permalink]  02 Jul 2011, 22:41
I had one question though .

Since Area of Traingle ABX = Area of Trianglr BXC

Will Area of triangle XYC = 1/2 Area of triangle ABC?

i.e Area of triangle XYC = 1/2 *64 = 32?

and will Area of triangle RCS = 1/4 (Area of triangle ABC)????

therefore, Area of triangle RCS = 1/4 *64 = 16??? Is this correct?

Can someone please verify if what I have done is correct?
Re: OG #12 Problem 109   [#permalink] 02 Jul 2011, 22:41
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