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Re: In right triangle triangle ABC, C is a right angle; DE is parallel to [#permalink]
Hi Bunuel
Do you have any plan to post the solution?
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Re: In right triangle triangle ABC, C is a right angle; DE is parallel to [#permalink]
KarishmaB Could you please explain this one? Thank you in advance!!
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Re: In right triangle triangle ABC, C is a right angle; DE is parallel to [#permalink]
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Bunuel wrote:

In right triangle \(\triangle ABC\), \(∠C\) is a right angle; \(\overline{DE}\) is parallel to \(\overline {AB}\); and D and E are the midpoints of \(\overline {CA}\) and \(\overline{ CB}\) , respectively. Each shaded triangle is similar to \(\triangle ABC\), and the shaded triangles are congruent to one another. What fraction of the area of \(\triangle ABC\) is shaded?

A. 1/8

B. 1/4

C. 1/3

D. 1/2

E. 3/4




This is a PS Butler Question



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When mid points of two sides are joined, the line is parallel to the third side and the smaller triangle obtained is similar to the large triangle.

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Screenshot 2023-04-03 at 10.44.56 AM.png
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So when D and E are joined, triangle CDE is similar to triangle CAB (sides and altitude ratios 1 : 2) so the ratio of their areas is 1 : 4.

Area of triangle CDE = (1/2) * CT * DE = x ....... (I)
So area of triangle ABC = 4x

Area of rectangle DEMP = DE * TN (DP and EM are perpendicular to the base because the shaded triangles are similar to triangle ABC)
Since TN = CT (because altitude ratios are 1 : 2 so CT : CN = 1:2)

Area of rectangle DEMP = DE * CT = 2x (from (I) above)

Since the shaded triangles are congruent, K is the mid point of PM and hence, area of shaded triangles is half the area of rectangle DEMP i.e. area of shaded triangles = x

Shaded area/ Area of ABC = x/4x = 1/4

Answer (B)
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In right triangle triangle ABC, C is a right angle; DE is parallel to [#permalink]
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https://gmatclub.com/forum/download/fil ... 33743fdeed

Sol. Draw a line on AB from C. let’s assume it as h.
Area of the triangle ABC =1/2×AB×h
Since shaded triangles are similar to the triangle ABC, hence shaded triangles are also right angle triangle. Height of the shaded triangle is ½ of the height ‘h’.
Combined base of both shaded triangles = PQ = DE = ½ of AB (as per midpoint theorem of triangle. )
Base of one shaded triangle = ¼ of AB
Area of both of the Shaded triangles =2×1/2×1/4 AB×1/2h=1/8×AB×h
Fraction of area shaded =((1/8×AB×h ))/((1/2×AB×h) )=1/4
Hence correct answer is B.
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In right triangle triangle ABC, C is a right angle; DE is parallel to [#permalink]
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