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Q.

In figure, LM ∥ NQ and LN ∥ PQ. If MP = $\frac{1}{3}$ MN , find the ratios of the areas of ΔLMN and ΔQNP.

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a

\frac{9}{4}

b

\frac{1}{2}

c

\frac{9}{2}

d

\frac{5}{2}

answer is A.

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Detailed Solution

Given LM ∥ NQ
LN ∥ PQ
MP =

\frac{1}{3}

MN −−−−−−−− (i)
Now since the line LM ∥ NQ, we can say
∠LMN = ∠ONP
We can say that the interior angles are equal as the two interior angles are cut by the transversal line MN so.
Now as the lines are parallel LN ∥ PQ, we can write
∠LNM = ∠OPN
By using the angle-angle criteria we can say
ΔLMN ∼ ΔNPQ [since ∠LMN = ∠ONP and ∠LNM = ∠OPN]
As the two triangles are congruent we can write the ratio of the area of the two triangles

\frac{Δ LMN}{ΔNPQ}

=

\frac{{MN}^{2}}{{NP}^{2}}

−−−−−−−− (ii)
It is given that
MP =

\frac{1}{3}

MN −−−−−−−− (i)
So we can write
⇒ NP = 1− MP
= 1 −

\frac{1}{3}

MN
=

\frac{2}{3}

MN
By substituting the value NP =

\frac{2}{3}

MN, in the equation (ii), we get
⇒

\frac{Δ LMN}{ΔNPQ}

=

\frac{{MN}^{2}}{{NP}^{2}}

=

\frac{{MN}^{2}}{{(\frac{2}{3} MN)}^{2}}

=

\frac{{MN}^{2}}{{\frac{4}{9} MN}^{2}}

Hence the ratios of the areas of ΔLMN and ΔQNP will be equal to

\frac{9}{4}

So, the ratios of the areas of ΔLMN and ΔQNP will be equal to

\frac{9}{4}

. The correct option is (1).

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