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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here. # In the figure, ABC and ADC are right triangles. Which of the  Question banks Downloads My Bookmarks Reviews Important topics
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Founder  Joined: 18 Apr 2015
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In the figure, ABC and ADC are right triangles. Which of the [#permalink]
Expert's post 00:00

Question Stats: 41% (02:11) correct 58% (03:05) wrong based on 12 sessions
Attachment: #GREpracticequestion In the figure, ABC and ADC are right triangle..jpg [ 16.2 KiB | Viewed 1127 times ]

In the figure, ABC and ADC are right triangles. Which of the following could be the lengths of AD and DC, respectively?

(I) $$\sqrt{3}$$ and $$\sqrt{4}$$
(II) 4 and 6
(III) 1 and $$\sqrt{24}$$
(IV) 1 and $$\sqrt{26}$$

(A) I and II only
(B) II and III only
(C) III and IV only
(D) IV and I only
(E) I, II and III only
[Reveal] Spoiler: OA

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Manager Joined: 04 Feb 2019
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Re: In the figure, ABC and ADC are right triangles. Which of the [#permalink]
Expert's post
We can use the Pythagorean theorem to find the length of AC:

$$3^{2} + 4^{2} = (AC)^{2}$$

$$9 + 16 = (AC)^{2}$$

$$25 = (AC)^{2}$$

$$5 = AC$$

We should also recognize this as a 3-4-5 triangle, one of the Pythagorean triplets.

In any case, we know that ADC is a right triangle, but we don't know which angle is the right angle. So, any of these three possibilities could be correct, depending on which side is the hypotenuse:

$$AD^{2} + DC^{2} = AC^{2}$$

$$AD^{2} + AC^{2} = DC^{2}$$

$$AC^{2} + DC^{2} = AD^{2}$$

Plug in AC = 5, and we get these possible equations:

$$AD^{2} + DC^{2} = 25$$

$$AD^{2} + 25 = DC^{2}$$

$$25 + DC^{2} = AD^{2}$$

I want to isolate the known value, 25:

$$25 = AD^{2} + DC^{2}$$

$$25 = DC^{2} - AD^{2}$$

$$25 = AD^{2} - DC^{2}$$

Now we can just plug in the answer choices. Clearly the first statement won't work, as 3 and 4 are much too small to add up to 25. Already that eliminates choices A, D, and E.

The second statement is pretty easy to eliminate as well, since 4² and 6² = 16 and 36, respectively. But no combination of adding or subtract 16 and 36 will give us an answer of 25. So Statement II must be false, and the answer must be C.
Intern Joined: 12 Nov 2018
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Re: In the figure, ABC and ADC are right triangles. Which of the [#permalink]
Can someone please explain C (especially IV option) is correct ? Thank you
Founder  Joined: 18 Apr 2015
Posts: 7401
Followers: 125

Kudos [?]: 1451 , given: 6611

Re: In the figure, ABC and ADC are right triangles. Which of the [#permalink]
Expert's post
In the case AC is the hypotenuse of the triangle, we have by The Pythagorean Theorem, $$AC^2 = AD^2 + DC^2$$

$$5^2 = AD^2 + DC^2$$

This equation is satisfied by III since $$5^2 = 1^2 + (\sqrt{24})^2$$ .

Therefore, III is possible.

Hope is more clear now to you Sir.

regards
_________________ GRE Instructor Joined: 10 Apr 2015
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Kudos [?]: 1986  , given: 20

Re: In the figure, ABC and ADC are right triangles. Which of the [#permalink]
1
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Expert's post
JelalHossain wrote:
Can someone please explain C (especially IV option) is correct ? Thank you

The trick here is to recognize that the right angle in triangle ACD could be in 3 different places.
∠CDA could be 90°
∠DCA could be 90°

Cheers,
Brent
_________________

Brent Hanneson – Creator of greenlighttestprep.com  Re: In the figure, ABC and ADC are right triangles. Which of the   [#permalink] 11 May 2019, 06:32
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