Tangents from an external point questions with answers

Questions

1. Question 1Angle at the centre

   In the diagram, $PA$ and $PB$ are tangents to the circle with centre $O$. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠OAP and ∠OBP$\angle OAP = 90^\circ$ (tan. ⊥ rad.)$\angle OBP = 90^\circ$ (tan. ⊥ rad.)
   2. Step 2: Angles in quadrilateral OAPB add up to 360°$\mathstrut x = 360^\circ - 90^\circ - 90^\circ - 54^\circ$ (∠ sum of quad.)$x = 126$
2. Question 2Isosceles triangle PAB

   In the diagram, $PA$ and $PB$ are tangents to the circle with centre $O$. Find $x$.

   Show the worked solution

   1. Step 1: Compare PA and PB$PA = PB$ (tangents from ext. pt.)So triangle PAB is isosceles.
   2. Step 2: Find x$\mathstrut x = \dfrac{180^\circ - 34^\circ}{2}$ (base ∠s of isos. △)$x = 73$
3. Question 3Find a length

   In the diagram, $PA$ and $PB$ are tangents to the circle with centre $O$. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠OAP$\angle OAP = 90^\circ$ (tan. ⊥ rad.)
   2. Step 2: Pythagoras' theorem in triangle OAP$OP^2 = OA^2 + PA^2$$\mathstrut x^2 = 12^2 + 16^2$$x^2 = 400$$x = \sqrt{400}$$x = 20$ cm
4. Question 4OP bisects the angles at P and at O

   In the diagram, $PA$ and $PB$ are tangents to the circle with centre $O$, and $OP$ is joined. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠OPA$\angle OPA = \angle OPB$ (tangents from ext. pt.)$\angle OPA = 19^\circ$
   2. Step 2: Find ∠OAP$\angle OAP = 90^\circ$ (tan. ⊥ rad.)
   3. Step 3: Angles in triangle OAP add up to 180°$x = 180^\circ - 90^\circ - 19^\circ$ (∠ sum of △)$x = 71$