P is a point outside a circle. From point P two rays are drawn to a circle which meet circle only at one point each. Let these 2 points be Q and R. The centre of circle is O. Which of the following statements is true and because of which property?
P is a point outside a circle. From point P two rays are drawn to a circle which meet circle only at one point each. Let these 2 points be Q and R. The centre of circle is O. Which of the following statements is true and because of which property?
The correct option is C PQ = PR, triangle congruency
Refer to above figure.
Line segments PQ and PR touch circle at only one point. Thus PQ and PR are tangents to circle.
Consider △ OQP and △ ORP,
Since Q and R lie on circumference, OR = OQ = radius.
[By Theorem- Tangents are perpendicular to radii at point of contact]
Thus tangents PQ and PR are perpendicular to corresponding radii OQ and OR respectively.
So, ∠ PQO = ∠ ROP = 90∘
Since OP is a common side between the 2 triangles, OP = OP = common side.
Thus by RHS congruency, triangles PQO and PRO are congruent.
Hence it is clear that PQ = PR since these 2 sides are corresponding sides in the 2 congruent triangles.
PQ and PR are nothing but the tangents from P to circle. This property will hold true for any pair of tangents drawn from an external point to a circle.
PQ = PR, triangle congruency
Refer to above figure.
Line segments PQ and PR touch circle at only one point. Thus PQ and PR are tangents to circle.
Consider △ OQP and △ ORP,
Since Q and R lie on circumference, OR = OQ = radius.
[By Theorem- Tangents are perpendicular to radii at point of contact]
Thus tangents PQ and PR are perpendicular to corresponding radii OQ and OR respectively.
So, ∠ PQO = ∠ ROP = 90∘
Since OP is a common side between the 2 triangles, OP = OP = common side.
Thus by RHS congruency, triangles PQO and PRO are congruent.
Hence it is clear that PQ = PR since these 2 sides are corresponding sides in the 2 congruent triangles.
PQ and PR are nothing but the tangents from P to circle. This property will hold true for any pair of tangents drawn from an external point to a circle.
