Ex.10.5 Q1 Circles Solution - NCERT Maths Class 9

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Question

In the given figure, \(A,\, B\) and \(C\) are three points on a circle with center \(O\) such that \(\begin{align}\angle {BOC}=30^{\circ} \text { and } \angle {AOB}=60^{\circ} \end {align}\)If  \(D\) is a point on the circle other than the arc \(ABC,\)  find \(\begin{align} \angle {ADC} \end{align}\)

 Video Solution
Circles
Ex 10.5 | Question 1

Text Solution

What is known?

Two angles subtended by arcs at the centre.

What is unknown?

Value of  \( \angle {ADC.}\)

Reasoning:

The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Steps:

\[\begin{align} \angle {AOC} &=\angle {AOB}+\angle {BOC} \\ &=90^\circ \end{align}\]

By Theorem \(10.8\) ,

\[\begin{align}  \angle {AOC} &=2 \angle {ADC}  \\ \angle ADC &= \frac{1}{2}\angle AOC\\\angle ADC &= \frac{1}{2} \times 90 = \,45^\circ \\\therefore \angle ADC &= \,45^\circ \end{align}\]
          

  
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