Question

Two circles intersects each other at point $A$ and $D$. Let the diameter $AB$ intersect the circle with centre $P$ in point $N$ and diameter $AC$ intersect the circle with centre of in the point $M$. then $AM.AC=AB.AN$

• A. True
• B. False

Answer 1

The correct option is A True

$AB$ and $AC$ are diameter of circle.

We know that, angle inscribed in semi-circle is ${90}^o$

$\therefore$ $\angle AMB=\angle ANC={90}^o$

In $△ACN$ and $△ABM,$

$\Rightarrow$ $\angle MAB=\angle NAC$ [ Vertically opposite angle ]

$\Rightarrow$ $\angle AMB=\angle ANC$ [ Angle in the semi-circle is a right angle ]

$\therefore$ $△ACN\sim △ABM$ [ By $AA$ similarity test ]

$\Rightarrow$ $\frac{AC}{AB}=\frac{AN}{AM}$

$\Rightarrow$ $AC\times AM=AB\times AN$