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 ∠ABC is 60∘. Find the measure of the angle OPN.
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 = 12 * 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∘. MOP = 90∘, and MNP = 45∘; MPO = PMO = 45∘. NPO = 180∘ – 75∘ – 45∘ – 45∘ = 15∘