Answer
Verified
481.2k+ views
Hint-Make use of the property of similar triangles and try to solve this problem
Using the data given let us draw the figure
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
$\begin{gathered}
\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
Using the data given let us draw the figure
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
$\begin{gathered}
\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
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE