Ex.10.5 Q10 Circles Solution - NCERT Maths Class 9

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Question

If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

 Video Solution
Circles
Ex 10.5 | Question 10

Text Solution

What is given ?

Two circles are drawn taking two sides of a triangle as diameters.

What is unknown?

To prove  that point of intersection of the \(2\) circles lie on the third side.

Reasoning:

Angle in a semicircle is a right angle. By using this fact we can show that \(BDC\) is a line which will lead to the proof that point of intersection lie on the third side.

Steps:

Since angle in a semicircle is a right angle, we get:

\( \begin{align} \angle  {ADB}&=90^{\circ} \text { and } \angle {ADC}=90^{\circ} \end{align}\)

\(\begin{align}  \angle {ADB}+\angle {ADC}&=90^{\circ}+90^{\circ} \\ \Rightarrow \;\; \angle {ADB}+\angle {ADC}&=180^{\circ} \end{align}\)

\(\Rightarrow \quad{BDC} \text { is a straight line. }\)

∴ \({D}\) lies on \({BC}\)

Hence, point of intersection of circles lie on the third side \(BC.\)

  
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