The side lengths of triangle ABC are such that AC BC AB.

Question

The side lengths of a right triangle ABC are such that AC > BC > AB. AC = 25 and AB = 9. What is the length of BC?

A. 16
B. \(4\sqrt{34}\)
C. 41
D. \(4\sqrt{52}\)
E. 256

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solution
Correct Answer: (B)

While this might appear to be a 3-4-5 triangle, you can’t square each side of a 3-4-5 and expect the pattern to hold: a 3-4-5 triangle must be some triangle whose sides can be reduced to the ratio 3x:4x:5x. As such, we have to resort to the Pythagorean Theorem, which in this case gives us 9^2+b^2=25^2, or b^2=544. At this point you can approximate – 544−−−√ is greater than 400−−−√, or 20, and less than 625−−−√, or 25, so the answer must be between 20 and 25: (B) the only such option.

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The query here isnt the answer to the problem but the solution given by veritas.. the way i deduced the answer was using the properties of triangle third side lies between the difference & sum of other two sides hence:
25-9<BC< 25+9
16<BC<34

and the only option falling is B. The explanation here says answer is between 20 and 25.. who is going wrong and where??

any help would be appreciated

Edited: Added 'right' to the question.

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Official Answer

B

Edited: Added 'right' to the question.

KarishmaB · Answer

I assume there is a diagram attached with it which gives some more information (that the triangle is a right triangle) etc. Also, judging from the question, we need to give the actual length of BC, not give the length that BC CAN take. Hence, there has to be some more info.

Given the limited info, your method is fine. Since 25 > BC > 9, the only possible options are (A) and (B). But sum of any two sides of a triangle must be greater than the third side so BC cannot be 16.
Answer (B)

swarman · Answer

I so agree with you, but surprisingly no diagram was attached with it.. Thank you for clarifying my approach though

VeritasPrepRon · Answer

I really like this question, because a lot of students immediately think the answer must be 16 to maintain that 3-4-5 pattern they've heard so much about, but obviously that is the trap answer for those going too fast. Once you've figured out that the Pythagorean Theorem will unlock the answer for you, the major hurdle is approximating square roots. This made me think of a blog I wrote on this topic a month or two back. It's actually perfect for exactly this question, so I figured I'd link it here in case it helped anyone: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2013/02/ron-point/ Thanks! -Ron

scottshrum · Answer

I agree... Great question, and great blog post, too!

SaraLotfy · Answer

I'm still having difficulty deciding between answer choices B and D

So I understand that the third side BC should be 16<BC<34

If I try to evaluate answer (B) 4sqt34 it gives 23.3 which is a possible answer. Also when I evaluate (D) 4sqt52 it gives 28.8 which is still a possible answer. I seem to be missing a trick.
Please help

Transcendentalist · Answer

There is a very simple way to solve this question. We know AC=25 , AB = 9 and assume BC = x

We also know that since it is a right triangle x^2 + 81 = 25^2

Now before embarking on lengthy calculations, it can easily be observed that for the equation x^2 should have the units digit as 4. By simple process of elimination and no lengthy calculations you can deduce that only option B fits the bill. (if x =4root34, x^2 units place = 4 since x^2 =16 x 34

Hope this makes sense

KarishmaB · Answer

Notice that the word 'right' had been added to the question. If it is a right triangle, you can easily use Pythagorean theorem and get your answer.

The original poster had tried to solve it for any generic triangle (the question without the word 'right'). 
You understand 16 < BC < 34.
Also, realize that you are given that AC > BC
Since AC = 25, BC must be less than 25. So only (B) works.

bumpbot · Answer

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The side lengths of triangle ABC are such that AC > BC > AB.

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