Question

In an isosceles triangle ABC, with AB = AC, the bisectors of $\angle B{\text{ and }}\angle {\text{C}}$ intersect each other at O. Join A to O. Show that : OB = OC

Answer 1

Hint: In this particular type of question we have to proceed by using the property of angle bisector at point B and C and then using the triangle property that angles opposite to equal sides of a triangle are equal and its converse is also true .

Complete step-by-step solution -

$\vartriangle ABC$ is an isosceles with AB = AC,
$\therefore \angle B = \angle C$
[ Since , angles opposite to equal sides are equal ]
$ \Rightarrow \dfrac{1}{2}\angle B = \dfrac{1}{2}\angle C$ [Divide both sides by 2]
$\begin{gathered}
   \Rightarrow \angle OBC = \angle OCB \\
  And{\text{ }}\angle {\text{OBA = }}\angle OCA \\
\end{gathered} $ [Angle bisectors]
$ \Rightarrow OB = OC$ [Side opposite to the equal angles are equal]

Note: Remember to recall the properties of triangle bisectors while solving this particular type of questions . Also constructing a diagram helps to get a clear picture of what we need to find out . Note that Angle Bisector Theorem says that an angle bisector of a triangle will divide the opposite side into two segments that are proportional to the other two sides of the triangle but the Angle Bisector Theorem Converse states that if a point is in the interior of an angle and equidistant from the sides, then it lies on the bisector of that angle .