Angle in a semicircle questions with answers

Questions

1. Question 1Find an angle

   In the diagram, $AB$ is a diameter of the circle with centre $O$, and $C$ lies on the circle. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠ACB$\angle ACB = 90^\circ$ (∠ in semicircle)
   2. Step 2: Angles in triangle ABC add up to 180°$\mathstrut x = 180^\circ - 90^\circ - 22^\circ$ (∠ sum of △)$x = 68$
2. Question 2Find a length (Pythagoras)

   In the diagram, $AB$ is a diameter of the circle with centre $O$, and $C$ lies on the circle. Find $x$, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Find ∠ACB$\angle ACB = 90^\circ$ (∠ in semicircle)
   2. Step 2: Pythagoras' theorem in triangle ABC$AB^2 = AC^2 + BC^2$$\mathstrut x^2 = 6^2 + 11^2$$x^2 = 157$$x = \sqrt{157}$$x = 12.5299\ldots$$\phantom{x} = 12.5$ cm (3 s.f.)
3. Question 3With an isosceles triangle

   In the diagram, $AB$ is a diameter of the circle with centre $O$, and $C$ lies on the circle. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠OBC$\angle OBC = 35^\circ$ (base ∠s of isos. △)
   2. Step 2: Find ∠ACB$\angle ACB = 90^\circ$ (∠ in semicircle)
   3. Step 3: Angles in triangle ABC add up to 180°$\mathstrut x = 180^\circ - 90^\circ - 35^\circ$ (∠ sum of △)$x = 55$
4. Question 4Does the circle on AB as diameter pass through C?

   In triangle $ABC$, angle $CAB = 28^\circ$ and angle $CBA = 62^\circ$. A circle is drawn with $AB$ as its diameter. Does the circle pass through $C$? Choose the answer with its reason.

   Show the worked solution

   1. Step 1: Find angle ACB (angle sum of triangle ABC)$\angle ACB = 180^\circ - 28^\circ - 62^\circ$$\phantom{\angle ACB} = 90^\circ$ (∠ sum of △)
   2. Step 2: C is on the circle with AB as diameter exactly when angle ACB = 90° (angle in a semicircle)Yes: angle ACB = 90°, so C lies on the circle with AB as diameter.