Question

in triangle PQR, MN // QR such that PM:Q M = 2:3 the ratio of the areas of triangle PQR and quadrilateral QRNM is

Answer 1

Here in ΔPQR, MN || QR

Hence

<PMN = <PQR ( Corresponding angles)

<PNM = <PRQ (Corresponding angles)

So by AA ΔPQR ~ ΔPMN

Hence

$\frac{Area of triangle PMN}{Area of triangle PQR}=(\frac{PM}{PQ})^{2} =(\frac{2}{5})^2=\frac{4}{25}$

So let area of ΔPMN = 4x

So area of ΔPQR = 25x

Hence area of quadrilateral QRNM = 25x - 4x = 21x

So

$\frac{Area of triangle PQR}{Area of quadrilateral QRNM}=\frac{25x}{21x}$

$\frac{Area of triangle PQR}{Area of quadrilateral QRNM}=\frac{25}{21}$

The ratio of the areas of triangle PQR and quadrilateral QRNM is 25 : 21

Answer 2

Here in ΔPQR, MN || QR

Hence

<PMN = <PQR ( Corresponding angles)

<PNM = <PRQ (Corresponding angles)

So by AA ΔPQR ~ ΔPMN

Hence

$\frac{Area of triangle PMN}{Area of triangle PQR}=(\frac{PM}{PQ})^2 =(\frac{2}{5} )^2=\frac{4}{25}$

So let area of ΔPMN $= 4x$

So area of ΔPQR$= 25x$

Hence area of quadrilateral QRNM $= 25x - 4x = 21x$

So

$\frac{Area of triangle PQR}{Area of triangle QRNM}=\frac{25x}{21x}$

$\frac{Area of triangle PQR}{Area of triangle QRNM}=\frac{25}{21}$

The ratio of the areas of triangle PQR and quadrilateral QRNM is $25 : 21$

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