Perpendicular bisector of a chord questions with answers

Questions

1. Question 1Find the radius

   In the diagram, $AB$ is a chord of the circle with centre $O$, and $OM$ is perpendicular to $AB$. $AB = 14$ cm. Find $x$, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Find AM$AM = MB$ (⊥ from centre bisects chord)$AM = 7$ cm
   2. Step 2: Pythagoras' theorem in triangle OAM$OA^2 = OM^2 + AM^2$$\mathstrut x^2 = 3^2 + 7^2$$x^2 = 58$$x = \sqrt{58}$$x = 7.6157\ldots$$\phantom{x} = 7.62$ cm (3 s.f.)
2. Question 2Find the distance from the centre

   In the diagram, $AB$ is a chord of the circle with centre $O$, and $OM$ is perpendicular to $AB$. $AB = 22$ cm. Find $x$, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Find AM$AM = MB$ (⊥ from centre bisects chord)$AM = 11$ cm
   2. Step 2: Pythagoras' theorem in triangle OAM$OM^2 = OA^2 - AM^2$$\mathstrut x^2 = 16^2 - 11^2$$x^2 = 135$$x = \sqrt{135}$$x = 11.6189\ldots$$\phantom{x} = 11.6$ cm (3 s.f.)
3. Question 3Find the length of the chord

   In the diagram, $AB$ is a chord of the circle with centre $O$, and $OM$ is perpendicular to $AB$. Find the length of the chord $AB$, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Pythagoras' theorem in triangle OAM$AM^2 = OA^2 - OM^2$$\mathstrut AM^2 = 12^2 - 8^2$$AM^2 = 80$$AM = 8.9442\ldots$ cm
   2. Step 2: Find AB$AB = 2 \times AM$ (⊥ from centre bisects chord)$AB = 17.8885\ldots$$\phantom{AB} = 17.9$ cm (3 s.f.)
4. Question 4Find an angle

   In the diagram, $AB$ is a chord of the circle with centre $O$, and $OM$ is perpendicular to $AB$. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠OBA, then x$\angle OBA = 22^\circ$ (base ∠s of isos. △)$\mathstrut x = 180^\circ - 2 \times 22^\circ$ (∠ sum of △)$x = 136$