Questions & Answers

Question

Answers

$\left( i \right)PR.BQ = QR.BP$

$\left( {ii} \right)AB \times CQ = BC \times AP$

Answer
Verified

Hint: - The problem can be solved easily using the Intercept theorem.

Given $\vartriangle ABC$ in which $BD$ is the bisector of$\angle B$ and a line $PQ\parallel AC$meets $AB,BC$ and $BD$ at $P,Q$ and $R$ respectively.

For proof \[\left( i \right)\]

Considering small $\vartriangle BQP,BR$ is the bisector of$\angle B$.

Using the properties of similar triangles we have

$

\therefore \dfrac{{BQ}}{{BP}} = \dfrac{{QR}}{{PR}} \\

\Rightarrow BQ.PR = BP.QR \\

\Rightarrow PR.BQ = QR.BP \\

$ (Rearranging the terms amongst themselves)

For proof \[\left( {ii} \right)\]

In$\vartriangle ABC$ we have

$PQ\parallel AC$ (Given in the question)

$

\Rightarrow \dfrac{{AB}}{{AP}} = \dfrac{{CB}}{{CQ}} \\

\Rightarrow AB \times CQ = CB \times AP \\

\Rightarrow AB \times CQ = BC \times AP \\

$ (By using intercept theorem and then rearranging the terms)

Note: - The intercept theorem, also known as Thaleâ€™s theorem or basic proportionality theorem, is an important theorem in elementary geometry about the ratios of various line segments that are created if two intersecting lines are intercepted by a pair of parallels. In order to use the Intercept theorem, the first and foremost thing is to recognize the triangle correctly.

Given $\vartriangle ABC$ in which $BD$ is the bisector of$\angle B$ and a line $PQ\parallel AC$meets $AB,BC$ and $BD$ at $P,Q$ and $R$ respectively.

For proof \[\left( i \right)\]

Considering small $\vartriangle BQP,BR$ is the bisector of$\angle B$.

Using the properties of similar triangles we have

$

\therefore \dfrac{{BQ}}{{BP}} = \dfrac{{QR}}{{PR}} \\

\Rightarrow BQ.PR = BP.QR \\

\Rightarrow PR.BQ = QR.BP \\

$ (Rearranging the terms amongst themselves)

For proof \[\left( {ii} \right)\]

In$\vartriangle ABC$ we have

$PQ\parallel AC$ (Given in the question)

$

\Rightarrow \dfrac{{AB}}{{AP}} = \dfrac{{CB}}{{CQ}} \\

\Rightarrow AB \times CQ = CB \times AP \\

\Rightarrow AB \times CQ = BC \times AP \\

$ (By using intercept theorem and then rearranging the terms)

Note: - The intercept theorem, also known as Thaleâ€™s theorem or basic proportionality theorem, is an important theorem in elementary geometry about the ratios of various line segments that are created if two intersecting lines are intercepted by a pair of parallels. In order to use the Intercept theorem, the first and foremost thing is to recognize the triangle correctly.

×

Sorry!, This page is not available for now to bookmark.