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Which of the following could be the area of an isosceles triangle with

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Which of the following could be the area of an isosceles triangle with [#permalink]

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Which of the following could be the area of an isosceles triangle with [#permalink]

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Bunuel wrote:
Which of the following could be the area of an isosceles triangle with perimeter 18 and one side of length 8?

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


If the triangle is isosceles, 2 sides are have equal lengths
We know that the perimeter of the triangle is 18,
2x + y = 18
where x is the length of the 2 sides with equal lengths
and y is the length of the other side

Now, we know that one of the sides is 8

There are two possibilities, when x = 8 or y =8
If x = 8, the three sides are 8,8,2
The area calculated by Heron's formula is not part of the list of the answer options

If y = 8, the three sides are 5,5,8
The area is \(2*\frac{1}{2}*4*3\) as it will form 2 right triangles with sides 3,4,5
Therefore, the area must be 12(Option B)
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Which of the following could be the area of an isosceles triangle with [#permalink]

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New post 01 Sep 2017, 01:19
pushpitkc wrote:
Bunuel wrote:
Which of the following could be the area of an isosceles triangle with perimeter 18 and one side of length 8?

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


If the triangle is isosceles, 2 sides are have equal lengths
We know that the perimeter of the triangle is 18,
2x + y = 18
where x is the length of the 2 sides with equal lengths
and y is the length of the other side

Now, we know that one of the sides is 8

There are two possibilities, when x = 8 or y =8
If x = 8, the three sides are 8,8,2 which is not possible(does not obey the properties of triangles)
If y = 8, the three sides are 5,5,8

The area is \(2*\frac{1}{2}*4*3\) as it will form 2 right triangles with sides 3,4,5
Therefore, the area must be 12(Option B)

Can u explain y 8,8,2 is not a valid triangle

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Re: Which of the following could be the area of an isosceles triangle with [#permalink]

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New post 01 Sep 2017, 01:31
Area of useless triangle is = a/4 x underroot of 4b2-a2 so it's and will be 12

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Which of the following could be the area of an isosceles triangle with [#permalink]

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New post 01 Sep 2017, 01:44
pushpitkc wrote:
Bunuel wrote:
Which of the following could be the area of an isosceles triangle with perimeter 18 and one side of length 8?

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


If the triangle is isosceles, 2 sides are have equal lengths
We know that the perimeter of the triangle is 18,
2x + y = 18
where x is the length of the 2 sides with equal lengths
and y is the length of the other side

Now, we know that one of the sides is 8

There are two possibilities, when x = 8 or y =8
If x = 8, the three sides are 8,8,2 which is not possible(does not obey the properties of triangles)
If y = 8, the three sides are 5,5,8

The area is \(2*\frac{1}{2}*4*3\) as it will form 2 right triangles with sides 3,4,5
Therefore, the area must be 12(Option B)


8,8, and 2 is a possible isosceles triangle of area ~8.
8-8=0 <2 , 8-2= 6 <8 , 8+8= 16> 2, 8+2=10>8.

it follows all required rules

Last edited by rahulkashyap on 01 Sep 2017, 01:50, edited 1 time in total.

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Re: Which of the following could be the area of an isosceles triangle with [#permalink]

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New post 01 Sep 2017, 01:49
Makesh617 wrote:
pushpitkc wrote:
Bunuel wrote:
Which of the following could be the area of an isosceles triangle with perimeter 18 and one side of length 8?

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


If the triangle is isosceles, 2 sides are have equal lengths
We know that the perimeter of the triangle is 18,
2x + y = 18
where x is the length of the 2 sides with equal lengths
and y is the length of the other side

Now, we know that one of the sides is 8

There are two possibilities, when x = 8 or y =8
If x = 8, the three sides are 8,8,2 which is not possible(does not obey the properties of triangles)
If y = 8, the three sides are 5,5,8

The area is \(2*\frac{1}{2}*4*3\) as it will form 2 right triangles with sides 3,4,5
Therefore, the area must be 12(Option B)

Can u explain y 8,8,2 is not a valid triangle

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Hi Makesh617,

My apologies, made the correction.
The area of a triangle with sides 8,8,2 is less than 8 and that is not an answer option
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Re: Which of the following could be the area of an isosceles triangle with [#permalink]

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New post 07 Sep 2017, 07:13
Bunuel wrote:
Which of the following could be the area of an isosceles triangle with perimeter 18 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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Re: Which of the following could be the area of an isosceles triangle with   [#permalink] 07 Sep 2017, 07:13
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