# Ex.7.2 Q2 Triangles Solution - NCERT Maths Class 9

## Question

In \(\Delta ABC, AD\) is the perpendicular bisector of \(BC\) (see the given figure). Show that \(\Delta ABC\) is an isosceles triangle in which \(AB = AC\).

## Text Solution

**What is Known?**

\(AD\) is perpendicular bisector of \(BC\) means and \(BD=DC\)

**To prove:**

\(\Delta ABC\) is an isosceles triangle in which \(AB = AC.\)

**Reasoning:**

We can show two triangles ADB and ADC congruent by using SAS congruency rule and then we can say corresponding parts of congruent triangles will be equal.

**Steps:**

In \(\Delta ADC\text{ and }\Delta ADB,\)

\(\begin{align}&AD=AD (Common)\\&\angle ADC=\angle ADB\ (\text{Each }\!\!~\!\!\text{ }{{90}^{{}^\circ }})\\&CD=BD\\&\left( \begin{array} & \text{AD is the perpendicular } \\ \text{bisector of BC} \\

\end{array} \right)\\\\&\therefore \Delta ADC\cong \Delta ADB \\&\text{(By SAS congruence rule)}\\\\&\therefore AB=AC\,\,\,(\text{By }CPCT)\!\!~\!\!\end{align}\)

Therefore, \(ABC\) is an isosceles triangle in which \(AB = AC\).