CH-18-001 · Two tangents: length and quadrilateral area

Diagram not accurately drawn

PA and PB are tangents to a circle with centre O. OA = 13 cm and OP = 42 cm. Calculate the total length PA + PB and the area of quadrilateral OAPB. Give both answers to 2 decimal places. Give reasons for the geometric facts you use.

Circle Theorems (Tangents from P) & Pythagoras Booklet p. 34, p. 85 & p. 86

Method

Write each calculation beside the geometric reason that justifies it

Worked Example (CH-18-001)

Maths	Reason
$\angle OAP=\angle OBP=90^\circ$	A radius is perpendicular to its tangent.
$PA=\sqrt{42^2-13^2}$	Pythagoras in right-angled triangle OAP.
$PB=PA\Rightarrow PA+PB=2\sqrt{1595}=79.87$ cm	Tangents from the same external point are equal.
$\text{Area}=2\times\frac12\times13\times\sqrt{1595}=519.19$ cm²	Add the areas of the two right-angled triangles.

Ans = PA + PB = 79.87 cm; area 519.19 cm².