# Ex.3.4 Q6 Understanding Quadrilaterals Solution - NCERT Maths Class 8

## Question

\(ABC\) is a right-angled triangle and \(O\) is the midpoint of the side opposite to the right angle. Explain why \(O\) is equidistant from \(A, B\) and \(C\). (The dotted lines are drawn additionally to help you).

## Text Solution

**What is Known?**

\(ABC\) is a right-angled triangle and \(O\) is the midpoint of the side opposite to the right angle.

**What is Unknown?**

Why \(O\) is equidistant from \(A, B\) and \(C\)

**Reasoning:**

Since, two right triangles make a rectangle and in any rectangle, diagonals bisect each other.

**Steps:**

\(ABCD\) is a rectangle as opposite sides are equal and parallel to each other and all the interior angles are of \(90^\circ .\)

\[\begin{align}{\rm{AD}}\left| {\left| {{\rm{BC}},{\rm{AB}}} \right|} \right|{\rm{DC}}\\{\rm{AD }} = {\rm{BC}},{\rm{ AB }} = {\rm{ DC}}\end{align}\]

In a rectangle, diagonals are of equal length and also these bisect each other.

Hence, \({\rm{AO }} = {\rm{ OC }} = {\rm{ BO }} = {\rm{ OD}}\)

Since, two right triangles make a rectangle where \(O\) is equidistant point from \(A, B, C\) and \(D\) because \(O\) is the mid-point of the two diagonals of a rectangle.

So, \(O\) is equidistant from \(A, B, C \) and \(D.\)