MathematicsIn the given figure, tangents AC and AB are drawn to a circle from a point A such that ∠BAC=30∘. A chord BD is drawn parallel to the tangent AC. Find ∠DBC.

In the given figure, tangents $AC$ and $AB$ are drawn to a circle from a point A such that $∠ BAC = 30^{\circ}$ . A chord $BD$ is drawn parallel to the tangent $AC$ . Find $∠ DBC$ .

A
$70^{\circ}$
B
$72^{\circ}$
C
$76^{\circ}$
D
$75^{\circ}$

Solution:

Given that the tangents AB and AC are drawn to a circle such that

∠ BAC = 30^{\circ}

.

We know that, from an external point, the length of tangents are equal.
So in

Δ A B C, A C = A B

.
Thus, by isosceles triangle property ,

∠ C = ∠ B

.
Apply angle sum property in

ΔABC

,

∠ A + ∠ B + ∠ C = 180^{\circ}

30^{\circ} + ∠ B + ∠ C = 180^{\circ}

30^{\circ} + 2 ∠ C = 180^{\circ}

2 ∠ C = 180^{\circ} - 30^{\circ}

2 ∠ C = 150^{\circ}

∠ C = 75^{\circ}

Given that

BD | | AC

.
So,

∠ ACB = ∠ CBD = 75^{\circ}

(alternate interior angles)
Hence, the value of

∠ DBC

is

75^{\circ}

.
Therefore, option 4 is correct.