Point $O$ lies in the interior of $\Delta PQR$. Given $O-A-P$,$O-B-Q$,$O-C-R$,$\overline{AB} \parallel \overline{PQ}$ and $\overline{BC} \parallel \overline{QR}$. Prove that $\overline{AC} \parallel \overline{PR}$.

Vedclass pdf generator app on play store
Vedclass iOS app on app store
(N/A) Given: Point $O$ lies in the interior of $\Delta PQR$. $O-A-P$,$O-B-Q$,$O-C-R$,$\overline{AB} \parallel \overline{PQ}$ and $\overline{BC} \parallel \overline{QR}$.
To prove: $\overline{AC} \parallel \overline{PR}$.
Proof:
In $\Delta OPQ$,since $\overline{AB} \parallel \overline{PQ}$,by the Basic Proportionality Theorem $(BPT)$,we have:
$\frac{OA}{AP} = \frac{OB}{BQ} \quad \dots (1)$
In $\Delta OQR$,since $\overline{BC} \parallel \overline{QR}$,by the Basic Proportionality Theorem $(BPT)$,we have:
$\frac{OB}{BQ} = \frac{OC}{CR} \quad \dots (2)$
From equations $(1)$ and $(2)$,we get:
$\frac{OA}{AP} = \frac{OC}{CR}$
In $\Delta OPR$,since $\frac{OA}{AP} = \frac{OC}{CR}$,by the converse of the Basic Proportionality Theorem $(BPT)$,we have:
$\overline{AC} \parallel \overline{PR}$.

Explore More

Similar Questions

In $\Delta ABC,$ the bisector of $\angle A$ intersects $\overline{BC}$ at $D.$ If $AC = 4.2,$ $DC = 3,$ and $BC = 5,$ find $AB.$

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM : AB = 5 : 7$ and $AN = 4$,find $NC$.

In $\Delta ABC$,$\overline{AD}$ is a median. Then,by Apollonius' theorem,$\ldots \ldots \ldots \ldots$ holds good.

In an equilateral triangle $ABC$,$D$ is a point on side $BC$ such that $BD = \frac{1}{3} BC$. Prove that $9 AD^{2} = 7 AB^{2}$.

Difficult
View Solution

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $\frac{AM}{MB} = \frac{2}{3}$ and $AC = 25$,find $NC$.

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial
For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free
For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo