Question

In the figure given below, ABOP is a rectangle and O is the centre of the circle. It is also given that AB = BC and the measure of the angle ABC is 60°. Find the measure of the angle OPN.

• A. 30°
• B. 15°
• C. 20°
• D. 25°
• E. 35°

Answer 1

The correct option is B 15°

AB = BC and ∠ABC=60°. Therefore, ΔABC is an equilateral triangle

Now see that ABOP is a rectangle.

And ∠BAN=60°, Therefore, ∠ NAP=90° − 60° = 30°

And ∠ANP = ½ * 90 = 45°

Now in ΔANP,

∠NPA=180°−45°−30°=105°

And hence <NPO = <NPA - <OPA = 105°- 90° = 15°

Shortcut

ABC is an equilateral triangle and ABM is a 30 – 60 -90 triangle (M being the point of intersection of AN and the circle). OMN is also 30^o. MOP = 90 ^o, and MNP = 45^o; MPO = PMO = 45^o. NPO = 180^o – 75^o – 45^o – 45^o = 15^o