In a circle, O is the center and AOB is the diameter. AT is a tangent to the circle. Line TB intersects the circle at Q. Given that ∠AOQ = 94°, find ∠ATQ.
Show answer & explanation
Correct answer: B
Explanation
The correct answer is option 2 i.e. 43°
AT is a tangent and ∠AOQ = 94°
The sum of the three angles of a triangle is 180°.
So, ∠BOQ = 180° - ∠AOQ = 180° - 94° = 86°
Also, ∠BAT = 90°
In ΔBOQ, OB = OQ
So, ∠OQB = ∠OBQ
∠OBQ + ∠OQB + ∠BOQ = 180°
⇒ 86° + 2∠OBQ = 180°
⇒ 2∠OBQ = 180° - 86° = 94°
⇒ ∠OBQ = 94°/2 = 47°
In ΔABT, ∠ABT + ∠BAT + ∠ATQ = 180°
⇒ ∠ATQ = 180° - (47° + 90°) = 180° - 137° = 43°
Therefore, the measure of ∠ATQ is 43°.