1) In the diagram, PQ and QR are tangents to the circle with centre O, at P and R R 20 S 50° Q Р respectively. Find the value of x.​

Answer:

Given, a circle with centre O and tangents QR and QP from point Q and points of contact to the circle R and P, respectively.

OR ⊥ QR (radius is perpendicular to tangent)

=> ∠ORQ = 90°

Also, OP ⊥ PQ (radius is perpendicular to tangent)

=> ∠OPQ = 90°

In quadrilateral ORQP,

∠OPQ + ∠PQR + ∠QRO + ∠ROP = 360° (angle sum property)

=> 90° + 50° + 90° + ∠ROP = 360°

=> ∠ROP = 360° - 230°

=> ∠ROP = 130°

Reflex ∠POR = 360° - 130° = 230° .... (1)

∠ROP = 2∠PSR (angle subtended at the centre of the circle is twice the angle subtended at any other point on the circle)

=> ∠PSR = [tex]\frac{130}{2}[/tex]

=> ∠PSR = 65°

In quadrilateral PORS,

∠POR + ∠ORS + ∠RSP + ∠SPO = 360° (angle sum property)

230° + 20° + 65° + ∠SPO = 360° (from (1))

∠SPO = 360° - 315°

∠SPO = 45°

x + ∠SPO = 90° (∵ OP ⊥ PQ)

X + 45° = 90°

X = 45°

∴ The value of x is 45°.

Explanation:

Given, a circle with centre O and tangents QR and QP from point Q and points of contact to the circle R and P, respectively.

OR ⊥ QR (radius is perpendicular to tangent)

=> ∠ORQ = 90°

Also, OP ⊥ PQ (radius is perpendicular to tangent)

=> ∠OPQ = 90°

In quadrilateral ORQP,

∠OPQ + ∠PQR + ∠QRO + ∠ROP = 360° (angle sum property)

=> 90° + 50° + 90° + ∠ROP = 360°

=> ∠ROP = 360° - 230°

=> ∠ROP = 130°

Reflex ∠POR = 360° - 130° = 230° .... (1)

∠ROP = 2∠PSR (angle subtended at the centre of the circle is twice the angle subtended at any other point on the circle)

=> ∠PSR = \frac{130}{2}

2

130

=> ∠PSR = 65°

In quadrilateral PORS,

∠POR + ∠ORS + ∠RSP + ∠SPO = 360° (angle sum property)

230° + 20° + 65° + ∠SPO = 360° (from (1))

∠SPO = 360° - 315°

∠SPO = 45°

x + ∠SPO = 90° (∵ OP ⊥ PQ)

X + 45° = 90°

X = 45°