its third side?

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sanju09: GMAT Instructor; Posts: 3650; Joined: Wed Jan 21, 2009 4:27 amFollowed by: 80 members; Location: India; Thanked: 267 times; GMAT Score:760

by sanju09 » Tue Mar 08, 2011 2:28 am

The longest side of a triangle is 20 and another of its side has length 10. If its area is 80, then what is the exact length of its third side?
(A) âˆš240
(B) âˆš250
(C) âˆš260
(D) âˆš270
(E) âˆš280

Made up

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Source: Beat The GMAT — Problem Solving | Tue Mar 08, 2011 2:28 am

anshumishra: Legendary Member; Posts: 543; Joined: Tue Jun 15, 2010 7:01 pmFollowed by: 3 members; Thanked: 147 times

by anshumishra » Tue Mar 08, 2011 4:06 am

sanju09 wrote:The longest side of a triangle is 20 and another of its side has length 10. If its area is 80, then what is the exact length of its third side?
(A) âˆš240
(B) âˆš250
(C) âˆš260
(D) âˆš270
(E) âˆš280

Made up

Lets say ABC is the triangle with sides BC = 20, AB = 10, AC = ?

Draw AD âŠ¥ BC, lets say AD = h

1/2 * h* 20 = 80 => h = 8

So, now in right angled triangle ABD, BD = 6, => BC = 20-6 = 14
Hence, in right angled triangle ADC , AC = âˆš[(14)^2 + (8)^2] = âˆš196+64 = âˆš260, C

Thanks
Anshu

(Every mistake is a lesson learned )

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ilikaroy: Junior | Next Rank: 30 Posts; Posts: 12; Joined: Thu May 13, 2010 10:08 pm; Thanked: 1 times

by ilikaroy » Tue Mar 08, 2011 7:58 pm

anshumishra wrote:
sanju09 wrote:The longest side of a triangle is 20 and another of its side has length 10. If its area is 80, then what is the exact length of its third side?
(A) âˆš240
(B) âˆš250
(C) âˆš260
(D) âˆš270
(E) âˆš280

Made up
Lets say ABC is the triangle with sides BC = 20, AB = 10, AC = ?

Draw AD âŠ¥ BC, lets say AD = h

1/2 * h* 20 = 80 => h = 8

So, now in right angled triangle ABD, BD = 6, => BC = 20-6 = 14
Hence, in right angled triangle ADC , AC = âˆš[(14)^2 + (8)^2] = âˆš196+64 = âˆš260, C

How are you getting BD = 6?

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anshumishra: Legendary Member; Posts: 543; Joined: Tue Jun 15, 2010 7:01 pmFollowed by: 3 members; Thanked: 147 times

by anshumishra » Tue Mar 08, 2011 8:07 pm

ilikaroy wrote:
anshumishra wrote:
sanju09 wrote:The longest side of a triangle is 20 and another of its side has length 10. If its area is 80, then what is the exact length of its third side?
(A) âˆš240
(B) âˆš250
(C) âˆš260
(D) âˆš270
(E) âˆš280

Made up
Lets say ABC is the triangle with sides BC = 20, AB = 10, AC = ?

Draw AD âŠ¥ BC, lets say AD = h

1/2 * h* 20 = 80 => h = 8

So, now in right angled triangle ABD, BD = 6, => BC = 20-6 = 14
Hence, in right angled triangle ADC , AC = âˆš[(14)^2 + (8)^2] = âˆš196+64 = âˆš260, C

How are you getting BD = 6?

Triangle ABD is also a right angled triangle; therefore ;
BD = âˆš[(AB)^2 - (AD)^2 = âˆš(10^2-8^2) = âˆš(100-64) = âˆš36 = 6.

Thanks
Anshu

(Every mistake is a lesson learned )

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Tue Mar 08, 2011 8:13 pm

One more approach:

If the sides of a triangle are a, b, and c, then the semi perimeter is given by s = (a+b+c)/2.
Also, the area is given by sqrt{s*(s-a)*(s-b)*(s-c)}.
If a = 20, b = 10, then s = (20+10+c)/2 = (30+c)/2.
So, 80 = sqrt{(30+c)/2 * (c-10)/2 * (c+10)/2 * (30 - c)/2}.
Or 6400*16 = (900 - c^2)*(c^2 - 100).
Note that going by options, c^2 is an integer.
Or (c^2-100) which is an integer, divides 16*6400.
Only c = sqrt(260), which gives c^2 - 100 as 160 will divide 16*6400.

The correct answer is hence sqrt(260).

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Night reader: Legendary Member; Posts: 1337; Joined: Sat Dec 27, 2008 6:29 pmFollowed by: 10 members; Thanked: 127 times

by Night reader » Wed Mar 09, 2011 2:25 am

Congrats Sanju, actually very worthy question
no way was for me to post yesterday, so posting now

several alternatives here ...

sanju09 wrote:The longest side of a triangle is 20 and another of its side has length 10. If its area is 80, then what is the exact length of its third side?
(A) âˆš240
(B) âˆš250
(C) âˆš260
(D) âˆš270
(E) âˆš280

Made up

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