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Re: D, E and F are points on the sides of triangle ABC above, such that qu [#permalink]

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04 Oct 2014, 12:41

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mvictor wrote:

D, E and F are points on the sides of triangle ABC above, such that quadrilateral AEFD is a rectangle. If DC=1, and CF=4, what is the value of AD?

(1) 3EB = AB

(2) FB = 2

It is very easy to solve if you mention two similar triangles.

Triangle CDF is similar to FEB since angle D=angle E=90 degrees angle DFC=angle EBF since DF||AB.

Because of similarity the corresponding sides of triangles must be proportional. Our task is to find such ratio.

(1) Sufficient. Here we know that proportional coefficient is 2, since DF=AE=AB-EB=3EB-EB=2EB. Each side in triangle CDF must be twice the corresponding side of triangle FEB. Therefore, AD=FE=1/2

(2) Sufficietnt CF=4, FB=2, hence proportional coefficient is 2. Each side in triangle CDF must be twice the corresponding side of triangle FEB. Therefore, AD=FE=1/2.

The correct answer is D. _________________

I'm happy, if I make math for you slightly clearer And yes, I like kudos:)

With data sufficiency questions, particularly geometry ones, you always want to make as many inferences as you can upfront. We can see that triangle CDF is a right triangle, because using the supplementary angle rule we know that angle CDF is 90 degrees. When they give us that leg DC = 1 and hypotenuse CF = 4 then, we can use the formula a^2 + b^2 = c^2 to see that leg DF = sqrt15. Because AEFD is a rectangle, AE must also be sqrt15. Statement 1 tells us that 3EB = AB, and therefore we know that EB = AB/2. Using the supplementary angle rule again we see that angle FEB is a 90 degree angle, and therefore triangle FEB is also a right triangle. Statement 2 gives us the hypotenuse of this triangle, and because we already have the information for leg EB (its sqrt15/2), we know that we can find the other leg EF. Again using the fact that AEFD is a rectangle, we know that EF will equal AD, and therefore with both statements together we can find AD, yielding answer choice C. We of course would not do the actual math, because all we want to prove is sufficiency.

Re: D, E and F are points on the sides of triangle ABC above, such that qu [#permalink]

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05 Jan 2018, 11:04

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