6. In the figure. O is the centre of a circle and AT is a tangent at point A.
The measure of angle BAT is:
A 30° B 60°
(D) 105°
(C) 75°
Answers
If O is the centre of a circle and AT is a tangent at point A, then the measure of angle BAT is option (B): 60°.
Step-by-step explanation:
It is given that,
O is the centre of the circle
AT is a tangent at point A
∠BCA = 60°
Step 1:
From the given figure we can see that AC is diameter and we know that an angle subtended by the diameter on any point of the circle is 90°
∴ ∠CBA = 90°
Consider ∆ABC, applying the Pythagoras theorem, we get
∠BCA + ∠CBA + ∠BAC = 180°
⇒ 60° + 90° + ∠BAC = 180°
⇒ ∠BAC = 180° - 150°
⇒ ∠BAC = 30° …. (i)
Step 2:
From the figure, we have OA as the radius and AT as the tangent.
We know that the tangent to a circle is perpendicular to the radius at the point of tangency.
∴ ∠OAT = 90° ….. (ii)
Now,
∠OAT = ∠BAC + ∠BAT
Substituting from (i) & (ii), we get
⇒ 90° = 30° + ∠BAT
⇒ ∠BAT = 90° - 30°
⇒ ∠BAT = 60°
Thus, the measure of the ∠BAT is 60°.
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