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# The length of one of the sides of a triangle is 13 units. If the area

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Retired Moderator
Joined: 27 May 2014
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The length of one of the sides of a triangle is 13 units. If the area  [#permalink]

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26 Feb 2018, 10:25
1
00:00

Difficulty:

85% (hard)

Question Stats:

45% (00:28) correct 55% (03:07) wrong based on 29 sessions

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The length of one of the sides of a triangle is 13 units. If the area of the triangle is 90 units^2 and the length of the another side of the triangle is 15 units. Find the length of the third side.

A) √124
B) √134
C) √224
D) √234
E) √244

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The length of one of the sides of a triangle is 13 units. If the area  [#permalink]

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26 Feb 2018, 10:53
1
saswata4s wrote:
The length of one of the sides of a triangle is 13 units. If the area of the triangle is 90 units^2 and the length of the another side of the triangle is 15 units. Find the length of the third side.

A) √124
B) √134
C) √224
D) √234
E) √244

Two Simple Approach for doing this Problem:

First Approach:

We have area of Triangle = $$\sqrt{s(s-a)(s-b)(s-c)}$$. Let c be the third side.
s = $$\frac{a+b+c}{2} = \frac{28+c}{2}$$
$$s(s-a)(s-b)(s-c) = (\frac{28+c}{2})(\frac{28-c}{2})(\frac{c+2}{2})(\frac{c-2}{2}) = 90*90 = 8100$$

$$c = \sqrt{244}$$

Second Approach:

Area of Triangle = $$\frac{1}{2}abSinC = \frac{1}{2}bcSinA = \frac{1}{2}acSinB$$
Let a,b be the side known to us i.e. a = 13, b =15
$$90 = \frac{1}{2}*13*15*SinC$$
$$SinC = \frac{12}{13}$$ or $$CosC = \frac{5}{13}$$

We have this Formula between angle and Sides of a Triangle

$$CosC = \frac{a^2 + b^2 - c^2}{2ab}$$
$$c^2 = 244$$ or $$c = \sqrt{244}$$
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The length of one of the sides of a triangle is 13 units. If the area  [#permalink]

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26 Feb 2018, 11:30
1
Formula used:
Area of the triangle = $$\sqrt{s(s-a)(s-b)(s-c)}$$ where semiperimeter s = $$\frac{a+b+c}{2}$$

If the third side is x, the semiperimter is $$\frac{13+15+x}{2} = \frac{28+x}{2}$$

$$\sqrt{s(s-a)(s-b)(s-c)}$$ = $$\sqrt{(\frac{28+x}{2})(\frac{28+x}{2} - 15)(\frac{28+x}{2} - 13)(\frac{28+x}{2} - x)}$$ = $$\sqrt{(\frac{28+x}{2})(\frac{x-2}{2})(\frac{x+2}{2})(\frac{28-x}{2})} = 90$$

Here, $$\sqrt{\frac{(28^2 - x^2)(x^2 - 4)}{16}} = 90$$ because $$a^2 - b^2 = (a-b)(a+b)$$

Squaring on both sides, $$(784 - x^2)(x^2 - 4) = 8100*16 = 16 * 81 * 100$$

A) x = √124 | (784 - 124)(124 - 4) = (660)(120)
B) x = √134 | (784 - 134)(134 - 4) = (650)(130)
C) x = √224 | (784 - 224)(224 - 4) = (560)(220)
D) x = √234 | (784 - 234)(234 - 4) = (550)(230)

E) x = √244 | (784 - 244)(244 - 4) = (540)(240) = (2*270)(3*80) = 16*81*100

Therefore, Option E(√244) is the length of the third side
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Joined: 06 Dec 2017
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GMAT 1: 730 Q49 V39
Re: The length of one of the sides of a triangle is 13 units. If the area  [#permalink]

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01 Mar 2018, 04:18
Is there no easier way to solve this question? Are we supposed to know this formula for the GMAT?
Director
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GMAT 1: 700 Q50 V33
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Re: The length of one of the sides of a triangle is 13 units. If the area  [#permalink]

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01 Mar 2018, 07:53
nuclearbattery wrote:
Is there no easier way to solve this question? Are we supposed to know this formula for the GMAT?

I couldn't find easier way to solve this... But Yes, these Formulas are of Great help.. !!

--== Message from the GMAT Club Team ==--

THERE IS LIKELY A BETTER DISCUSSION OF THIS EXACT QUESTION.
This discussion does not meet community quality standards. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.

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Re: The length of one of the sides of a triangle is 13 units. If the area &nbs [#permalink] 01 Mar 2018, 07:53
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