# Ex.7.1 Q3 Triangles Solution - NCERT Maths Class 9

## Question

\(AD\) and \(BC\) are equal perpendiculars to a line segment \(AB\) (See the given figure). Show that \(CD\) bisects \(AB\).

## Text Solution

**What is Known?**

\({\text{AD}} \bot {\text{AB, BC}} \bot {\text{AB}}\;{\text{and AD}} = {\text{BC}}\)

**To prove:**

\(CD\) bisects \(AB\) or \(OA = OB\)

**Reasoning:**

We can show two triangles \(OBC\) and \(OAD\) congruent by using AAS congruency rule and then we can say corresponding parts of congruent triangles will be equal.

**Steps:**

In \(\Delta BOC\) and \(\Delta AOD\),

\[\begin{align}\angle BOC& = \angle AOD\\\text{(Vertically opp}&\text{osite angles)}\\\\\angle CBO &= \angle DAO\text{(Each}\, 90^ {\circ})\\BC &= AD \text{(Given)}\\\therefore \Delta BOC &\cong \Delta AOD\\ \text{(AAS} \,& \text{congruence rule)}\\\\\therefore BO &= AO \text{(By CPCT)}\end{align}\]

Hence \(CD\) bisects \(AB\).