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A rhombus PQRS has A, B, C, D as mid points of its sides PQ, QR, RS an

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A rhombus PQRS has A, B, C, D as mid points of its sides PQ, QR, RS an  [#permalink]

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New post 15 Aug 2018, 10:21
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A rhombus PQRS has A, B, C, D as mid points of its sides PQ, QR, RS and SP respectively. A, B, C, D are joined to form quadrilateral ABCD. What is the perimeter of quadrilateral ABCD?

(1) PQ = 10 units.

(2) PR = 16 units.
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A rhombus PQRS has A, B, C, D as mid points of its sides PQ, QR, RS an  [#permalink]

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New post 15 Aug 2018, 11:36
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amanvermagmat wrote:
A rhombus PQRS has A, B, C, D as mid points of its sides PQ, QR, RS and SP respectively. A, B, C, D are joined to form quadrilateral ABCD. What is the perimeter of quadrilateral ABCD?

(1) PQ = 10 units.

(2) PR = 16 units.


ABCD is a rectangle. (When we joint the midpoints of a rhombus , a rectangle emerges)

Rephrased question stem:- What is the perimeter of the rectangle ABCD?
a) Length & breadth , both required

St1:- PQ = 10 units
We can only determine the diagonals of the rectangle. PQ=BD=10=PS=AC.
Not sufficient to determine length & breadth of the rectangle.

St2:- PR = 16 units

In the triangle QRP, AB || PR and A & B are the midpoints of two of it's sides , so \(AB=\frac{PR}{2}=\frac{16}{2}\)=8 units=breadth of rectangle.

There is no sufficient info to determine the length of rectangle, BC.

Combining, we have, in the right angled triangle ABC, AC=diagonal=10 units(from st1) and AB=8 units (from st2)

So, \(BC=\sqrt{10^2-8^2}\)=6 units

Now perimeter can be determined using the formula: P=2(AB+BC)
Sufficient.

Ans. (C)
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A rhombus PQRS has A, B, C, D as mid points of its sides PQ, QR, RS an &nbs [#permalink] 15 Aug 2018, 11:36
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