Class X Session 2023-24 Question 21

 MATH SAMPLE QUESTION PAPER

Class X Session 2023-24

MATHEMATICS STANDARD (Code No.041)

SECTION B - Question 21

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.

Explanation :

ABCD is a parallelogram.

we know that AB = DC

let AB = l

then AB = DC = l

given that Point P divides AB in the ratio 2:3

$$\Rightarrow AP=\frac{2}{5}\times l;andBP=\frac{3}{5}\times l$$

$$SinceweconsideredAP+BP=AB=l$$

$$\Rightarrow DQ=\frac{4}{5}\times l;andCQ=\frac{1}{5}\times l$$

given that point Q divides DC in the ratio 4:1

$$SinceDQ+CQ=DC=l$$

As we know that [AA similarity]

$$\therefore ∆APO\sim ∆CQO$$

$$\therefore \frac{AP}{CQ}=\frac{PO}{QO}=\frac{AO}{CO}$$

$$\therefore \frac{AO}{CO}=\frac{\frac{2}{5}\times l}{\frac{1}{5}\times l}=\frac{2}{1}$$

After simplifying the above equation, we get

$$\Rightarrow OC=\frac{1}{2}\times AO$$