• Quick retrieval from earlier learning (Circle theorems; Pythagoras' theorem). Students answer on mini whiteboards.
• Show answers together and note who needs support in the main task.

1. A tangent meets a radius at the point of contact. A triangle formed with the centre has an angle of 45° at the centre. What is the angle between the tangent and the line to the centre (the third angle)? → 45Prompt: A tangent is perpendicular to the radius at the point of contact.
2. Two angles at the circumference stand on the same arc. One is 48°. What is the other? → 48Prompt: Angles in the same segment are equal.
3. A triangle is drawn in a semicircle with one side the diameter. One of the other angles is 27°. What is the third angle (not the right angle)? → 63Prompt: The angle in a semicircle is 90°.

• Show a photo of a building, a tile pattern or a road sign and ask what shapes and angles students can spot.
• Share the objective: Circle theorems: tangents and radii. Read the success criteria together.

• Work through the example out loud, saying why you do each step, not just what you do.
• Example: PA and PB are tangents from P to a circle with centre O, touching it at A and B. ∠APB = 49°. Find ∠AOB.
• Step 1: ∠OAP = ∠OBP = 90°
• Step 2: ∠AOB = 360° − 90° − 90° − 49° = 131°
• Answer: 131
• Pause after each step and ask a student to say what comes next.

• Do these together. Students try each step on whiteboards before you reveal it.
• Use the hints as prompts rather than giving the answer.

1. PA and PB are tangents from P to a circle with centre O, touching it at A and B. ∠APB = 32°. Find ∠AOB. → 148Prompt: A tangent meets the radius at 90°.
2. Two tangents are drawn from point P to a circle, touching it at A and B. PA = (4x + 4) cm and PB = (5x + 1) cm. Find x. → 3Prompt: Tangents from the same point are equal in length.

• Students work on the student sheet. Circulate and check the first two answers of every student early.
• Students who finish: write a question of their own for a partner, with the answer.

The 8 questions are on the student sheet below, with answers in the key.

• Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.

1. A tangent from P touches a circle with centre O at T. The radius is 16 cm and OP = 34 cm. Find the length PT. → 30Prompt: Triangle OTP has a right angle at T.
2. PA and PB are tangents from P to a circle with centre O, touching it at A and B. ∠APB = 31°. Find ∠AOB. → 149Prompt: A tangent meets the radius at 90°.

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-10.