Tangent perpendicular to radius questions with answers

Questions

1. Question 1Find a length (Pythagoras)

   In the diagram, $PA$ is a tangent to the circle with centre $O$ at $A$. Find $x$, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Find ∠OAP$\angle OAP = 90^\circ$ (tan. ⊥ rad.)
   2. Step 2: Pythagoras' theorem in triangle OAP$PA^2 = OP^2 - OA^2$$\mathstrut x^2 = 8^2 - 3^2$$x^2 = 55$$x = \sqrt{55}$$x = 7.4161\ldots$$\phantom{x} = 7.42$ cm (3 s.f.)
2. Question 2Find an angle

   In the diagram, $PA$ is a tangent to the circle with centre $O$ at $A$. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠OAP$\angle OAP = 90^\circ$ (tan. ⊥ rad.)
   2. Step 2: Angles in triangle OAP add up to 180°$\mathstrut x = 180^\circ - 90^\circ - 46^\circ$ (∠ sum of △)$x = 44$
3. Question 3Tangent and a chord

   In the diagram, $AT$ is a tangent to the circle with centre $O$ at $A$, and $AB$ is a chord. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠OAT, then ∠OAB$\angle OAT = 90^\circ$ (tan. ⊥ rad.)$\mathstrut \angle OAB = 90^\circ - 57^\circ$$\angle OAB = 33^\circ$
   2. Step 2: Find ∠OBA, then x$\angle OBA = 33^\circ$ (base ∠s of isos. △)$\mathstrut x = 180^\circ - 2 \times 33^\circ$ (∠ sum of △)$x = 114$
4. Question 4Is the line a tangent?

   $A$ is a point on the circle with centre $O$ and radius $3$ cm. Is $AP$ a tangent to the circle at $A$? Choose the answer with its reason.

   Show the worked solution

   1. Step 1: Compare OA² + AP² with OP²$OA^2 + AP^2 = 3^2 + 4^2$$\phantom{OA^2 + AP^2} = 25$$OP^2 = 5^2$$\phantom{OP^2} = 25$
   2. Step 2: Is angle OAP a right angle? (converse of Pythagoras' theorem)$OA^2 + AP^2 = OP^2$, so $\angle OAP = 90^\circ$.
   3. Step 3: AP is a tangent at A exactly when it is perpendicular to the radius OAYes: angle OAP = 90°, so AP is perpendicular to the radius OA.