Circles · Learner pages

Lesson 1 · Locate the radius perpendicular to a tangent

Identify a tangent and apply the right angle at its contact point.

Name or initials: ____________   Date: ____________

A tangent touches a circle at one point; a secant intersects it at two points.

1. Identify the contact. The line touches the circle at T. Join the centre O to T to draw a radius.
2. Use the theorem. The radius OT is perpendicular to the tangent at T, so the contact angle is 90°.
3. Understand why the foot matters. The perpendicular from O to a line is its shortest distance. If a different point on a tangent were closer than T, it would lie inside the circle and the line would cut the circle.
4. Use the right angle in problems. When an external point P lies on the tangent, △OTP is right-angled at T. Pythagoras and triangle-angle rules apply.

Worked example: A circle has radius 5 cm and OP = 13 cm for an external point P with tangent PT. Since OT ⟂ PT, PT = √(13² − 5²) = √144 = 12 cm.

1. A line touches a circle at T, whose centre is O. What angle does OT make with that tangent?

   1. 90° · 2. 45° · 3. 180°

2. OT = 5 cm is a radius, PT is tangent at T, and OP = 13 cm. Find PT.

   1. 8 cm · 2. 18 cm · 3. 12 cm

3. A straight line meets a circle at two distinct points. What is it called?

   1. A radius · 2. A secant · 3. A tangent

Discuss: Draw a tangent at the top of a circle. Show which radius creates the right angle and distinguish a parallel line that misses the circle.

Lesson 2 · Compare tangents from one external point

Use equal tangent lengths and the congruent-triangle reason.

Name or initials: ____________   Date: ____________

Radii of the same circle have equal lengths.

1. Name both contacts. PA and PB are tangents from the same outside point P. Join O to A, O to B and O to P.
2. Find equal parts. OA = OB because they are radii. OP is common, and both contact angles are 90°.
3. Apply RHS congruence. Right triangles OAP and OBP are congruent by hypotenuse OP and equal radius sides. Therefore PA = PB.
4. Use the equality. If PA = 2x + 1 and PB = x + 6, then 2x + 1 = x + 6, so x = 5 and both tangent lengths are 11.

Worked example: If PA = 3x − 2 and PB = x + 8, equality gives 2x = 10 and x = 5. Both lengths equal 13 units, which are positive as required.

1. From external point P, tangents PA and PB touch the same circle. If PA = 4 cm, find PB.

   1. 4 cm · 2. 8 cm · 3. 5 cm

2. From external point P, tangents PA and PB touch the same circle. If PA = 7 cm, find PB.

   1. 14 cm · 2. 8 cm · 3. 7 cm

3. From external point P, tangents PA and PB touch the same circle. If PA = 9 cm, find PB.

   1. 10 cm · 2. 9 cm · 3. 18 cm

Discuss: Sketch two different outside points. Explain why the theorem pairs two tangent lengths at each point, not all four lengths together.

Lesson 3 · Connect central and tangent angles

Combine two tangent right angles with an angle sum.

Name or initials: ____________   Date: ____________

The interior angles of a quadrilateral total 360°.

1. Add both radii. For tangents PA and PB, join OA and OB. This creates quadrilateral OAPB.
2. Mark the contact angles. ∠OAP = 90° and ∠OBP = 90°. Together they contribute 180°.
3. Use the quadrilateral sum. Thus ∠AOB + ∠APB = 180° when AOB is the non-reflex angle inside this quadrilateral.
4. Calculate a concrete angle. If ∠AOB = 120°, then ∠APB = 180° − 120° = 60°. Check all four angles add to 360°.

Worked example: If the angle between two tangents is 50°, then the relevant central angle is 180° − 50° = 130°. The four interior angles are 90°, 90°, 50° and 130°.

1. PA and PB are tangents from P. The non-reflex central angle AOB is 80°. Find angle APB.

   1. 100° · 2. 80° · 3. 280°

2. PA and PB are tangents from P. The non-reflex central angle AOB is 90°. Find angle APB.

   1. 45° · 2. 270° · 3. 90°

3. PA and PB are tangents from P. The non-reflex central angle AOB is 100°. Find angle APB.

   1. 260° · 2. 80° · 3. 100°

Discuss: Choose three possible central angles between 0° and 180°. Calculate the tangent angle and describe how the two change together.

Final check · Set 1

Try one set after the lessons. Use the other set after reviewing any errors.

1. OT = 3 cm, OP = 5 cm and PT is tangent at T. Find PT.

   2 cm / 8 cm / 4 cm

2. In right triangle OTP formed by centre O, contact T and external point P, which side is the hypotenuse?

   PT / OP / OT

3. From external point P, tangents PA and PB touch the same circle. If PA = 15 cm, find PB.

   30 cm / 16 cm / 15 cm

4. From external point P, tangents PA and PB touch the same circle. If PA = 18 cm, find PB.

   19 cm / 18 cm / 36 cm

5. PA and PB are tangents from P. The non-reflex central angle AOB is 120°. Find angle APB.

   120° / 240° / 60°

6. PA and PB are tangents from P. The non-reflex central angle AOB is 130°. Find angle APB.

   230° / 50° / 130°

Final check · Set 2

Try one set after the lessons. Use the other set after reviewing any errors.

1. A tangent segment PT is 15 cm and radius OT is 8 cm. Find OP.

   17 cm / 7 cm / 23 cm

2. A tangent segment PT is 8 cm and radius OT is 6 cm. Find OP.

   2 cm / 14 cm / 10 cm

3. From external point P, tangents PA and PB touch the same circle. If PA = 20 cm, find PB.

   20 cm / 40 cm / 21 cm

4. From external point P, tangents PA and PB touch the same circle. If PA = 23 cm, find PB.

   46 cm / 24 cm / 23 cm

5. PA and PB are tangents from P. The non-reflex central angle AOB is 140°. Find angle APB.

   40° / 140° / 220°

6. PA and PB are tangents from P. The non-reflex central angle AOB is 150°. Find angle APB.

   150° / 210° / 30°