Question

$\angle BAC$ of triangle $ABC$ is obtuse and $AB=AC$. $P$ is a point in $BC$ such that $PC=12$ cm. $PQ$ and $PR$ are perpendiculars to sides $AB$ and $AC$ respectively. If $PQ=15$ cm and $=9$ cm; find the length of $PB$.

• A. $20$
• B. $24$
• C. $36$
• D. $18$

Answer 1

The correct option is A $20$

Given: $AB=AC$, $PQ\perp AB$ and $PR\perp AC$
Since, $AB=AC$
$\angle ABC=\angle ACB$...(I) (Isosceles triangle property)
Now, In $△PBQ$ and $△PRC$
$\angle PBQ=\angle PCR$ (From I)
$\angle PQB=\angle PRC$ (Each ${90}^{\circ }$)
$\angle QPB=\angle RPC$ (Third angle)

Thus, $△QPB\sim △RPC$ (by AAA similarity)

If two triangles are similar, then their corresponding sides are proportional.
$\frac{PQ}{PR}=\frac{PB}{PC}$
$\frac{15}{9}=\frac{PB}{12}$
$PB=\frac{15\times 12}{9}$
$PB=20cm$