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The short diagonal of a rhombus is three-quarters the length of the

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The short diagonal of a rhombus is three-quarters the length of the  [#permalink]

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New post 16 Feb 2018, 02:45
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The short diagonal of a rhombus is three-quarters the length of the long diagonal. If the perimeter of the rhombus is 20, then what is the area of the rhombus?

A 14
B 15
C 24
D 25
E 40

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Re: The short diagonal of a rhombus is three-quarters the length of the  [#permalink]

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New post 16 Feb 2018, 02:56
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Tridhipal wrote:
The short diagonal of a rhombus is three-quarters the length of the long diagonal. If the perimeter of the rhombus is 20, then what is the area of the rhombus?

A 14
B 15
C 24
D 25
E 40


Perimeter = 20

so the sides of rhombus are 5 each. Each sides forms a triangle i.e 3,4,5 triplet.

So (3,4,5) , (3,4,5) , (3,4,5) , (3,4,5)

D1 = 3+3
D2 = 4+4

area = 1/2 D1 D2 = 1/2 x 6 x 8 = 24

(C) imo
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Re: The short diagonal of a rhombus is three-quarters the length of the  [#permalink]

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New post 16 Feb 2018, 05:00
1
Tridhipal wrote:
The short diagonal of a rhombus is three-quarters the length of the long diagonal. If the perimeter of the rhombus is 20, then what is the area of the rhombus?

A 14
B 15
C 24
D 25
E 40


You are given that perimeter is 20 and you know that each side of a rhombus is equal. So each side would be 5.
Now knowing that the hypotenuse is 5 doesn't necessarily mean that the other two sides will be 3 and 4. They could very well be 2 and \(\sqrt{21}\) or infinite other values.
But there is one more data point - the short diagonal is (3/4)th the length of the long diagonal. So the two are in the ratio 3:4. This means the ratio of half of their lengths (the legs of the right triangle) will also be 3:4. So we know that the two legs will be 3 and 4 giving the hypotenuse as 5.

Area of rhombus = 4* Area of triangle = 4 * (1/2)*3*4 = 24

Answer (C)
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Re: The short diagonal of a rhombus is three-quarters the length of the  [#permalink]

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New post 23 Dec 2018, 01:50
Basic principles to make clear.

A rhombus is a subset of parallelograms. The diagonals bisect each other. The resulting 4 triangles are right angle triangles, with sides forming a hypotenuse, and side lengths of 3-4-5, because a rhombus always has perpendicular diagonals (the technical term for a quadrilateral that always has right angle bisectors is 'orthodiagonal quadrilateral').
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Re: The short diagonal of a rhombus is three-quarters the length of the   [#permalink] 23 Dec 2018, 01:50
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