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its third side?

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) √240
(B) √250
(C) √260
(D) √270
(E) √280


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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) √240
(B) √250
(C) √260
(D) √270
(E) √280


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

Draw AD ⊥ 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 = √[(14)^2 + (8)^2] = √196+64 = √260, C

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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) √240
(B) √250
(C) √260
(D) √270
(E) √280


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

Draw AD ⊥ 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 = √[(14)^2 + (8)^2] = √196+64 = √260, C

Image
How are you getting BD = 6?
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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) √240
(B) √250
(C) √260
(D) √270
(E) √280


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

Draw AD ⊥ 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 = √[(14)^2 + (8)^2] = √196+64 = √260, C

Image
How are you getting BD = 6?
Triangle ABD is also a right angled triangle; therefore ;
BD = √[(AB)^2 - (AD)^2 = √(10^2-8^2) = √(100-64) = √36 = 6.
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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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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) √240
(B) √250
(C) √260
(D) √270
(E) √280


Made up
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