Question

In the following figure, PQ = QR, ∠RQP  = 68°, PC and CQ are tangents to the circle with centre O.

Calculate the values of:

1. ∠QOP
2. ∠QCP

Answer 1

In the figure, PQ = QR ∠RQP = 68°

PC and QC are tangents to the circle with centre O from C.

In ∠PQR,

PQ = QR   ...(Given)

∴ ∠PRQ = ∠RPQ

But ∠PRQ + ∠RPQ + ∠RQP = 180°  ...(Sum of angles of a triangle)

`\implies` ∠PRQ + ∠PRQ + 68° = 180°

`\implies` 2∠PRQ = 180° – 68° = 112°

∴ `∠PRQ = 112^circ/2 = 56^circ`

Now QC is tangent and PQ is chord

`\implies` ∠PQC = ∠PRQ = 56°

But ∠PQC = ∠QPC  ...(∵ PC = QC tangents from C)

∴ ∠QPC = 56°

In ΔPQC,

∠C + ∠PQC + ∠QPC = 180°  ...(Angles of a triangle)

∠C + 56° + 56° = 180°

`\implies` ∠C + 112° = 180°

`\implies` ∠C = 180° – 112° = 68°

But ∠POQ + ∠C = 180°

∴ ∠POQ + 68° = 180°

∴ ∠POQ = 180° – 68° = 112°

Hence ∠QOP = 112° and ∠QCP = 68°