Ex.9.4 Q6 Areas of Parallelograms and Triangles Solution - NCERT Maths Class 9

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Question

Diagonals \(AC \) and \(BD\) of a quadrilateral \(ABCD\) intersect each other at \(P\). Show that

\(\begin{align}&{\rm{ ar }}\left( {\Delta APB} \right) \times {\rm{ ar }}\left( {\Delta CPD} \right) \\ &= {\rm{ ar }}\left( {\Delta APD} \right) \times {\rm{ ar }}\left( {\Delta BPC}\right)\end{align}\)

[Hint: From \(A\) and \(C\), draw perpendiculars to \(BD\)]

 Video Solution
Areas Of Parallelograms And Triangles
Ex 9.4 | Question 6

Text Solution

What is known?

A quadrilateral ABCD, in which diagonals AC and BD intersect each other at point P.

What is unknown?

To Prove:

\(\begin{align}&{\rm{ ar }}\left( {\Delta APB} \right) \times {\rm{ ar }}\left( {\Delta CPD} \right) \\ &= {\rm{ ar }}\left( {\Delta APD} \right) \times {\rm{ ar }}\left( {\Delta BPC}\right)\end{align}\)

Reasoning:

From A, draw \(AM\bot BD\) and from C, draw \(CN\bot BD.\) Now we can find area of triangles to get the required result.

Steps:

Proof :

\(\begin{align} \text{ ar} (\Delta ABP) & \!\!=  \!\frac{1}{2} \!\!\times PB \!\times \! AM...\text{(i)} \\ \text{ar} (\Delta APD) & \!\!=\!\frac{1}{2} \!\times \! PD \! \times \! AM...\text{(ii)} \end{align} \)

Dividing eq. (ii) by (i), we get,

\(\begin{align}  \frac{\text{ar} (\Delta \text{APD)}}{\text{ar} (\Delta ABP)} & =\frac{\frac{1}{2} \times PD\times AM}{\frac{1}{2}\times PB\times AM} \\ \Rightarrow \frac{\text{ar}(\Delta APD)}{\text{ar}(\Delta ABP)}&=\frac{PD}{PB}.....(\text{iii}) \\ \end{align}\)

Similarly,

\(\begin{align} & \frac{\text{ar}(\Delta CDP)}{\text{ar}(\Delta BPC)}=\frac{PD}{PB}....(\text{iv}) \end{align} \)

From eq . (iii) and (iv), we get

\(\begin{align} & \frac{\text{ar}(\Delta APD)}{\text{ar}(\Delta ABP)}=\frac{\text{ar}(\Delta CDP)}{\text{ar} (\Delta BPC)} \\ & \Rightarrow \text{ar} (\Delta APD)\times \text{ar}(\Delta BPC) \\ & \quad =\text{ar}(\Delta ABP) \times \text{ar} (\Delta CDP) \end{align}\)

Hence proved.

 Video Solution
Areas Of Parallelograms And Triangles
Ex 9.4 | Question 6