Question

BOA is a diameter of a circle and the tangent at a point P meets BA extended at T. If ∠PBO = 30°, then ∠PTA is equal to 30°. State ‘True’ or ‘False’​

Answer 1

Answer:

$\large\bf\underline{question}$

BOA is a diameter of a circle and tangent at a point P meets BA extended at T.if ∠PBO = 30°,then ∠PTA is equal to 30°.state true or false.

$\large\bf\underline{given}$

• diameter of the circle = BOA

• ∠PBO = 30°

• PT is tangent

$\large\bf\underline{to\:find}$

wheather the statement is true or false

$\large\bf\underline{solution}$

Given,PT is tangent

∠PBA = 30°

so,

∠BPA = 90° (angle in semicircle)

∠OPT = 90° (angle between the radius and tangent)

also in ∆BPA

∠PAB = 180°- (∠30°+ ∠BPA)

⠀⠀⠀⠀= 180° - 30° - 90°

⠀⠀⠀⠀= 60° ⠀⠀⠀⠀⠀⠀↦ (1)

Now look into the ∆OPA

OP = OA (radius of circle)

➩∠OAP = ∠OPA = 60°

( ∵ ∠OAP = ∠ BAP = 60°)

∴ ∠APT = ∠OPT - ∠OPA

⠀⠀⠀⠀⠀= 90° - 60° = 30°

∴∠PAT = 180° - (60°) = 120°

(supplimentary angle)

in ∆PAT

∠PAT + ∠PTA + ∠APT = 180°

➔ 120° + ∠PTA + 30° = 180°

➔∠PTA = 30°

$\large\bf\underline{answer}$

$\tt\underline\pink{∠PBO\:=\:∠PTA\:,hence\:the\: statement\:is\:true}$