Name: Proof of Basic Proportionality theorem
Uploaded: 2019-07-19
Duration: 3 min 10 s
Description: When a line is drawn parallel to one side, it intersects the other two sides of a triangle in the same ratio. Watch this video to learn more about them.

Question

$A B C D$ is a parallelogram. Point $P$ divides $A B$ in the ratio $2 : 3$ and point $Q$ divides $D C$ in the ratio $4 : 1$ . Prove that $O C$ is half of $O A$ .

Accepted Answer

Given $A B C D$ is a parallelogram such that $A P : P B = 2 : 3$ and $D Q : C Q = 4 : 1$ or $C Q : Q D = 1 : 4$ .
To prove $C O = \frac{1}{2} A O$ .

Proof: $A B C D$ is a parallelogram, So, opposite sides are parallel and equal, i.e. $A B ‖ C D$ and $A D ‖ B C .$

In $Δ A P O$ and $Δ C Q O$ , we have,

$∠ A O P = ∠ C O Q$ [They are vertically opposite angles. Hence, they are equal]

$∠ O P A = ∠ O Q C$ [ $A B ‖ C D$ and $P Q$ is transversal. So, they are alternate interior angles. Hence, they are equal]

$∴ Δ A P O ~ Δ C Q O$ [by AA similarity criterion]

Then, $\frac{A P}{C Q} = \frac{P O}{Q O} = \frac{A O}{C O} - - - - (i)$ [The corresponding sides of similar triangles are proportional]

Given, $A P = \frac{2}{5} A B - - - - (i i)$ .

$[∵ A P : P B = 2 : 3 \Rightarrow A P = \frac{2}{5} A B]$

and, $C Q = \frac{1}{5} C D = \frac{1}{5} A B - - - - (i i i)$

$[∵ C Q : Q D = 1 : 4 \Rightarrow C Q = \frac{1}{5} C D and C D = A B]$

On dividing Eq. (ii) by Eq. (iii), we get, $\frac{A P}{C Q} = \frac{\frac{2}{5} A B}{\frac{1}{5} A B} = \frac{2}{1}$

$\Rightarrow \frac{A O}{C O} = \frac{2}{1}$ [From equ (i)]

$\Rightarrow C O = \frac{1}{2} A O$ .

Hence, proved.

$A B C D$ is a parallelogram. Point $P$ divides $A B$ in the ratio $2 : 3$ and point $Q$ divides $D C$ in the ratio $4 : 1$ . Prove that $O C$ is half of $O A$ .

Important Questions on Triangles

Learn and Prepare for any exam you want !

ABCD is a parallelogram. Point P divides AB in the ratio 2:3 and point Q divides DC in the ratio 4:1. Prove that OC is half of OA.

Important Questions on Triangles

Learn and Prepare for any exam you want !

$A B C D$ is a parallelogram. Point $P$ divides $A B$ in the ratio $2 : 3$ and point $Q$ divides $D C$ in the ratio $4 : 1$ . Prove that $O C$ is half of $O A$ .