Question

In the given figure, if $RB,SC$ and $TD$ are perpendiculars to $BD$ with lengths $x,z$ and $y$, then the ratio of $x$ and $y$ is :

• A. $\frac{p}{q}$
• B. $\frac{p}{p+q}$
• C. $\frac{q}{p}$
• D. $\frac{q}{q+p}$

Answer 1

The correct option is A $\frac{p}{q}$

Given: $RB,SC$ and $TD$ are three perpendiculars to the same base $BD$.

Hence, $RB\parallel SC\parallel TD$

By theorem on equal intercepts, we get

$\frac{BC}{CD}=\frac{RS}{SD}=\frac{BS}{ST}$

$\therefore \frac{p}{q}=\frac{RS}{SD}=\frac{BS}{ST}$ ...(1)

In $△RSB$ and $△DST$,

$\frac{RS}{SD}=\frac{BS}{ST}$ ....From (1)

$\angle RSB=\angle DST$ ...Vertically opposite angles

$△RSB\sim △DST$ ....S.A.S test of similarity

$\frac{RS}{SD}=\frac{BS}{ST}=\frac{RB}{TD}$ ...(2)

$\therefore \frac{x}{y}=\frac{p}{q}$ ...From (1) and (2)