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Length of an isoscles triangle

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by gmattesttaker2 » Sat Feb 22, 2014 10:55 am
Hello,

Can you please assist with this:

Which of the following could be the area of an isosceles triangle with perimeter l8 and one side
of length 8 ?

(A) 6
(B) 12
(C) 14
(D) 16
(E) 18

OA: B

I tried to solve as follows:

Since in any triangle the third side must be less than the sum of the other 2 sides we can have here:

5 + 5 > 8 - Case 1
8 + 2 > 8 - Case 2

In both these cases the perimeter is 18. However, I think in Case 2, the angle opposite 2 would be 90 degrees and since it is greatest angle in this triangle it cannot face 2.

So I am thinking that the sides of the triangle are 5, 5 and 8. However I am not sure if this is correct since it doesn't look right when I plug in the values for the 45:45:90 angles.

Can you please assist?

Thanks,
Sri
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by Brent@GMATPrepNow » Sat Feb 22, 2014 11:05 am
gmattesttaker2 wrote:Hello,

Can you please assist with this:

Which of the following could be the area of an isosceles triangle with perimeter l8 and one side
of length 8 ?

(A) 6
(B) 12
(C) 14
(D) 16
(E) 18
Hey Sri,

Don't worry about this question. It requires us to use something called Heron's formula (sometimes called Hero's formula). More here: https://www.mathsisfun.com/geometry/herons-formula.html

Knowledge of this formula is NOT REQUIRED on the GMAT.

What's the source of this question?

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by gmattesttaker2 » Sat Feb 22, 2014 11:36 am
Brent@GMATPrepNow wrote:
gmattesttaker2 wrote:Hello,

Can you please assist with this:

Which of the following could be the area of an isosceles triangle with perimeter l8 and one side
of length 8 ?

(A) 6
(B) 12
(C) 14
(D) 16
(E) 18
Hey Sri,

Don't worry about this question. It requires us to use something called Heron's formula (sometimes called Hero's formula). More here: https://www.mathsisfun.com/geometry/herons-formula.html

Knowledge of this formula is NOT REQUIRED on the GMAT.

What's the source of this question?

Cheers,
Brent
Hello Brent,

Thank you very much for the clarification. This question is a part of a question set that I got for practice problems. Unfortunately they don't have the detailed solutions. Just the answer. Thanks for all your help.

Best Regards,
Sri
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by Scott@TargetTestPrep » Wed Jan 03, 2018 7:20 am
gmattesttaker2 wrote:Which of the following could be the area of an isosceles triangle with perimeter l8 and one side of length 8 ?

(A) 6
(B) 12
(C) 14
(D) 16
(E) 18
Since the isosceles triangle has a perimeter of 18 and a side length of 8, we could have the following sides:

1) 8, 8, 2

or

2) 8, 5, 5

In option 1, the base is 2 and the legs (the sides that have equal length) are 8 each. The height of this triangle, h, satisfies the Pythagorean theorem in the form (b/2)^2 + h^2 = l^2 where b is the base and l is a leg of the isosceles triangle. Thus:

(2/2)^2 + h^2 = 8^2

1 + h^2 = 64

h^2 = 63

h = √63

Recall that the area of a triangle is (b x h)/2, so the area of the triangle is (2 x √63)/2 = √63. However, this is not one of the answer choices. Thus, we must consider option 2.

In option 2, the base is 8 and the legs are 5 each. So, we have:

(b/2)^2 + h^2 = l^2

(8/2)^2 + h^2 = 5^2

16 + h^2 = 25

h^2 = 9

h = √9 = 3

So, the area of the triangle is (8 x 3)/2 = 24/2 = 12.

Answer: B

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