Question

PA and PB are tangent lines to the circle whose center is O. If ∠APO = 30°then ∠AOB = ?

​

Answer 1

Answer:

∠ APO =30

∘

.....given

From P we have two tangents PA and PB

We know that if we join point P and center of circle

O then the line PO divides the angle between tangents

⇒∠ APO = ∠ OPB = 30

∘

.....(i)

∠ OAP = ∠ OBP = 90

∘

....... radius is perpendicular to tangent ....(ii)

Consider quadrilateral OAPB

⇒∠ OAP + ∠ APB + ∠ PBO + ∠ AOB = 360

∘

....sum of angles of quadrilateral

From figure ∠ APB = ∠ APO + ∠ OPB

⇒∠ OAP + ∠ APO + ∠ OPB + ∠ PBO + ∠ AOB = 360

∘

Using (i) and (ii)

⇒90

∘

+30

∘

+30

∘

+90

∘

+∠ AOB = 360

∘

⇒240

∘

+∠ AOB = 360

∘

⇒∠ AOB = 120

∘

Hence ∠ AOB is 120

∘

Answer 2

$\huge\boxed{\fcolorbox{red}{aqua}{♥️Hola♥️}}$

∠ APO =30°

Given

From P we have two tangents PA and PB

We know that if we join point P and center of circle

O then the line PO divides the angle between tangents

⇒∠ APO = ∠ OPB = 30° .....(i)

∠ OAP = ∠ OBP = 90°

radius is perpendicular to tangent ....(ii)

Consider quadrilateral OAPB

⇒∠ OAP + ∠ APB + ∠ PBO + ∠ AOB = 360°

Sum of angles of quadrilateral

From figure ∠ APB = ∠ APO + ∠ OPB

⇒∠ OAP + ∠ APO + ∠ OPB + ∠ PBO + ∠ AOB = 360°

Using (i) and (ii)

⇒ 90° +30° +30° +90° +∠ AOB = 360°

⇒240° +∠ AOB = 360°

⇒∠ AOB = 120°

Hence ∠ AOB is 120°