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Let ABC be a triangle and M be a point on side AC closer to vertex C than A. Let N be a point on side AB such that MN is parallel to BC and let P be a point on side BC such that MP is parallel to AB. If the area of the quadrilateral BNMP is equal to \[\frac{5}{{18}}\] times the area of triangle ABC, then the ratio AM/MC equals.
A. 5
B. 6

Last updated date: 15th Jul 2024
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Hint-Make use of the property of similar triangles and try to solve this problem
Using the data given let us draw the figure

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Let us consider the length of AM to be=x and MC =y
Also it is given that MN is parallel to BC, so BN is the transversal
So from this we get $\begin{gathered}
  \angle ANM = \angle ABC \\
  \angle AMN = \angle ACB \\
\end{gathered} $ (corresponding angles)
And also MP is parallel to NB.
So, we get $\angle ANM = \angle MPC$(Since MP is parallel to BN )
So, from this we can write $\vartriangle ANM \sim \vartriangle MPC \sim \vartriangle ABC$
So by theorem, ratio of areas of two similar triangles is equal to the ratio of squares of their corresponding sides.
$\frac{{area\vartriangle ANM}}{{area\vartriangle ABC}} = \frac{{{{(AM)}^2}}}{{{{(AC)}^2}}} = \frac{{{x^2}}}{{{{(x + y)}^2}}}$ ------(1)
$\frac{{area\vartriangle MPC}}{{area\vartriangle ABC}} = \frac{{{{(MC)}^2}}}{{{{(AC)}^2}}} = \frac{{{y^2}}}{{{{(x + y)}^2}}}$----------(2)
From the data it is given that $area\vartriangle ANC + area\vartriangle MPC = area\vartriangle ABC - area\square NMCB = area\vartriangle ABC - \dfrac{5}{{18}}\vartriangle ABC = \dfrac{{13}}{{18}}area\vartriangle ABC$ Now ,let us add eq(1) and eq(2)
So we get
  \frac{{13}}{{18}} = \frac{{{x^2} + {y^2}}}{{{{(x + y)}^2}}} \\
   \Rightarrow 5{x^2} - 26xy + 5{y^2} = 0 \\
   \Rightarrow 5{x^2} - 25xy - xy + 5{y^2} = 0 \\
   \Rightarrow (5x - 1)(x - 5y) = 0 \\
   \Rightarrow \frac{x}{y} = 5{\text{ OR }}\frac{x}{y} = \frac{1}{5} \\
\end{gathered} $
But, also it is given that x>y
Therefore the answer is 5
Option A is the correct answer
Note: Modify the equation and bring it to a simplified form based on the data which is given in the question