In the picture, a triangle is formed by two mutually perpendicular tangents to a circle and a third tangent.Prove that the perimeter of the triangle is equal to the diameter of the circle.
Answers
Perimeter of Triangle = Diameter of circle if a triangle is formed by two mutually perpendicular Tangents & a third Tangent
Step-by-step explanation:
in ΔOMP & ΔOPN
OM = ON = R
OP = OP = Common
PM = PN ( Equal Tangents)
ΔOMP ≅ ΔOPN
∠MPO = ∠NPO
∠MPN = 90° ( given)
=> ∠MPO = ∠NPO = 45°
=> ∠MOP = ∠NOP = 45° ( also as ∠QMP = ∠QNP = 90°)
=> ∠MPO = ∠MOP
& ∠NPO = ∠NOP
=> OM = MP & ON = NP
=> MP = NP = R
MP = PQ + QM
NP = PR + RN
=> MP + NP = R + R
=> PQ + QM + PR + RN = 2R
QM = QS ( Equal tangent)
RN = RS ( Equal Tangent)
=> PQ + QS + PR + RS = 2R
QS + RS = QR
=> PQ + QR + PR = 2R
Perimeter of Triangle = Diameter of circle
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