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.
the measure of the angle OPN.
Answer: Option B
:
B
AB = BC and∠ABC=60°. Therefore, ΔABC is an equilateral triangle
Now see thatABOP 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∘
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B
AB = BC and∠ABC=60°. Therefore, ΔABC is an equilateral triangle
Now see thatABOP 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∘
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