Question

O is any point on the bisector of the acute angle $\angle$XYZ. From O, a line is extended to join XY such that OP is parallel to ZY. Then, $△$YPO is:

• A. Scalene
• B. Isosceles but not right angled
• C. Equilateral
• D. Right & isosceles

Answer 1

The correct option is B

Isosceles but not right angled

$\angle$ POY = $\angle$ OYZ (alternate angles)

$\angle$Y is bisected, so $\angle$ POY = $\angle$ PYO
Hence, PY = PO
So, $△$YPO is isosceles

Also, it is given that $\angle$XYZ is acute, so any angle which is half of it (bisected by OY) is less than ${45}^{\circ }$.
or $\angle$ PYO + $\angle$OYZ < ${90}^{\circ }$

Hence, the third angle of the $△$YPO i.e. $\angle$ YPO will be obtuse to satisfy angle-sum property of a triangle.

Hence, $△$YPO is isosceles but not right angled.