Last visit was: 11 Sep 2024, 03:55 It is currently 11 Sep 2024, 03:55
Toolkit
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

# In right triangle triangle ABC, C is a right angle; DE is parallel to

SORT BY:
Tags:
Show Tags
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 95450
Own Kudos [?]: 657530 [9]
Given Kudos: 87241
Tutor
Joined: 11 May 2022
Posts: 1080
Own Kudos [?]: 759 [2]
Given Kudos: 81
Intern
Joined: 28 Mar 2022
Posts: 4
Own Kudos [?]: 0 [0]
Given Kudos: 210
Intern
Joined: 31 Jan 2016
Posts: 1
Own Kudos [?]: [0]
Given Kudos: 4
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?
Intern
Joined: 01 Nov 2022
Posts: 16
Own Kudos [?]: 4 [0]
Given Kudos: 33
GPA: 3.16
Re: In right triangle triangle ABC, C is a right angle; DE is parallel to [#permalink]
Tutor
Joined: 16 Oct 2010
Posts: 15296
Own Kudos [?]: 67970 [1]
Given Kudos: 442
Location: Pune, India
Re: In right triangle triangle ABC, C is a right angle; DE is parallel to [#permalink]
1
Kudos
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

Check the links to other Butler Projects:

Attachment:
The attachment 1 (1).jpg is no longer available

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.

Attachment:

Screenshot 2023-04-03 at 10.44.56 AM.png [ 20.76 KiB | Viewed 1394 times ]

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

Intern
Joined: 05 Sep 2022
Posts: 12
Own Kudos [?]: 7 [1]
Given Kudos: 1
Location: India
Schools: Wharton '25
In right triangle triangle ABC, C is a right angle; DE is parallel to [#permalink]
1
Bookmarks

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