Question

In the given figure, if ABC is the tangent to a circle at B whose centre is O .PQ is a chord parallel to AC and
$\bold{ \angle \: QBC = 65 \: \degree}$
,then find the value of
$\bold\red{\angle \: PBQ }$

Either give correct answer or else id will banned permanently .
No sorry or request in my question as it is not your personal app ​

Answer 1

Given:-

• ABC is the tangent to a circle at B whose centre is O.
• PQ is a chord parallel to AC
• ∠QBC = 65°

To Find:-

• ∠PBQ.

Solution:-

ABC is the tangent to the circle at B.

In the given figure we can see that:-

OB ⊥ AC

Also,

OM ⊥ PQ

We know,

• When a line is perpendicular to chord, it bisects the chord.

Hence,

• PM = PQ
• ∠PMB = ∠QMB
• PB = QB

Hence by SAS criteria

∆PMB ≅ ∆QMB

By this we can say,

• ∠PBM = ∠QBM

We know,

• A tangent to a circle is always perpendicular to the radius of the circle

Hence,

∠PBM = ∠QBM = 90° - 65° = 25°

∴ ∠PBM + ∠QBM = ∠PBQ

⇒ 25° + 25° = ∠PBQ

⇒ ∠PBQ = 50°

∴ ∠PBQ measures 50°.

________________________________

Extra Informations:-

• The lengths of tangents drawn from external point to a circle are always equal.

Definition:-

Tangent:- A tangent to a circle is a line that intersects a circle at only one point.

________________________________

Answer 2

As OQ⊥ PR

(radius ⊥ tangent) & AB∣∣PR

⇒AB⊥OL

and perpendicular from centre to chord bisects the chord.

⇒AL=BL,

∠QLB=∠QLA

& LQ=LQ\Rightarro$$ By SAS

congruency, ΔQLA≅ΔQLB

⇒∠AQL=∠BQL=20o (90o−70o)

⇒∠AQB=40o.