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In the figure /PX/ = /XQ/, PQ//YZ and XV//QR. What is the ra...

$In the figure /PX/ = /XQ/, PQ//YZ and XV//QR. What is the ratio of the area of XYZQ to te area of \(\bigtriangleup\)YZR?$

In the figure /PX/ = /XQ/, PQ//YZ and XV//QR. What is the ratio of the area of XYZQ to te area of $\bigtriangleup$ YZR?

A.

1:2

B.

2:1

C.

1:2

D.

3:1

View answer & explanation

Correct answer is B

From the diagram, XYZQ is a parallelogram. Thus, |YZ| = |XQ| = |PX|; $\bigtriangleup$ PXY, Let the area of XYZQ = A₁, the area of $\bigtriangleup$ PXY

= Area of $\bigtriangleup$ YZR = A₂

Area of $\bigtriangleup$ PQR = A = A₁ + 2A₂

But from similarity of triangles

$\frac{\text{Area of PQR}}{\text{Area of PXY}} = (\frac{PQ}{PX})^2 = (\frac{QR}{XY})^2$

$\frac{A}{A_2} = (\frac{2}{1})^2 = \frac{2}{1}$

A = 4A₂ But, A = A₁ + 2A

A₁ = 4A₂ - 2A₂

A₁ = 2A₂

$\frac{A_1}{A_2}$ = 2

A₁:A₂ = 2:1

Area of XYZQ:Area of $\bigtriangleup$ YZR = 2:1

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