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