GCSE Maths: Tangents and radii (Higher)
Part of the GCSE Maths circle theorems chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Tangents and radii
A tangent is a straight line that touches a circle at exactly one point, called the point of contact. It never goes inside the circle. There are two important facts about tangents.
Fact 1: a tangent is perpendicular to the radius at the point of contact. The angle between a tangent and the radius that meets it is 90°. Reason to write: “the angle between a tangent and a radius is 90°”. Why? Every point on the tangent except the point of contact is outside the circle, so it is further from the centre O than a radius. So the radius OT is the shortest distance from O to the tangent line. The shortest distance from a point to a straight line is always the perpendicular one. So OT is perpendicular to the tangent.
Fact 2: tangents from an external point are equal in length. If you draw two tangents TA and TB to a circle from a point T outside it, then TA = TB. Reason to write: “tangents from an external point are equal in length”. Why? Triangles OAT and OBT are right-angled (Fact 1) and share the hypotenuse OT. By Pythagoras, TA2 = OT2 − OA2 and TB2 = OT2 − OB2. Since OA = OB (radii) the two lengths are equal.
The kite. The quadrilateral OATB is a kite. Its two right angles at A and B mean the other two angles add up to 180°: angle AOB + angle ATB = 180°. The line OT is the line of symmetry. It bisects angle ATB and angle AOB, and it is perpendicular to the chord AB. Triangle TAB is isosceles (because TA = TB), and triangle OAB is isosceles (because OA = OB).
Key ideas
- A tangent is perpendicular to the radius at the point of contact. Reason: “the angle between a tangent and a radius is 90°”.
- Tangents from an external point are equal in length. Reason: “tangents from an external point are equal in length”.
- The right angle makes Pythagoras' theorem available: (tangent length)2 + (radius)2 = (distance from the centre)2.
- Quadrilateral OATB is a kite: angle AOB + angle ATB = 180°.
- Triangles TAB and OAB are both isosceles.
Watch out
The right angle is between the tangent and the radius, at the point of contact. It is not at the external point T, and it is not between the two tangents.Key words. A tangent is a straight line that touches a circle at exactly one point, the point of contact.
Method. A tangent is at right angles to the radius at the point where it touches the circle. That makes triangle OAT right-angled.
- Draw the triangle. Join O to A and to T. The radius OA, the tangent TA and the line OT make triangle OAT.triangle OAT
- Find the angle at the point of contact. A tangent touches the circle at one point and is perpendicular to the radius there.Angle OAT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Use the angle sum of triangle OAT. The three angles add up to 180°. Subtract the two we know.Angle AOT = 180° − 90° − 30° = 60° (the angles in a triangle add up to 180°)
Check. 60° + 30° + 90° = 180°. ✓
Watch out. The right angle is at A, the point of contact, between the radius and the tangent.
Method. The two radii and the two tangents make a quadrilateral OATB with two right angles. Use the angle sum 360°.
- Draw the radii OA and OB. They go to the two points of contact and make a quadrilateral OATB with the two tangents.quadrilateral OATB
- Mark the right angles. Each tangent is perpendicular to the radius drawn to its point of contact.Angle OAT = 90° and angle OBT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Use the quadrilateral OATB. Its four angles add up to 360°. Subtract the three we know.Angle AOB = 360° − 90° − 90° − 64° = 116° (the angles in a quadrilateral add up to 360°)
Check. 360° = 360°. ✓ A shortcut for later: angle AOB + angle ATB = 180°.
Watch out. The quadrilateral OATB is a kite. Its two right angles are at A and B, opposite each other.
Method. Two tangents from one point are equal in length, so triangle TAB is isosceles.
- Show triangle TAB is isosceles. Tangents drawn from the same point to a circle are equal in length.TA = TB (tangents from an external point are equal in length)
- Write the base angles. The base angles of an isosceles triangle are equal: the base angles are at A and B.Angle TAB = angle TBA
- Use the angle sum. Subtract the angle at T and share the rest equally.Angle TAB = (180° − 50°) ÷ 2 = 130° ÷ 2 = 65° (the angles in a triangle add up to 180°)
Check. 65° + 65° + 50° = 180°. ✓
Watch out. Say why the triangle is isosceles: the tangents from T are equal. Do not just say "it looks isosceles".
Method. The radius meets the tangent at a right angle, so triangle OAT is right-angled and we can use Pythagoras' theorem.
- Find the right angle. A tangent is perpendicular to the radius at the point of contact.Angle OAT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Write Pythagoras for triangle OAT. The longest side is OT, opposite the right angle.OT2 = OA2 + TA2
132 = 52 + TA2 - Rearrange. Take the known square away from both sides.TA2 = 169 − 25 = 144
- Take the square root. A length is positive.TA = √144 = 12 cm
Check. 52 + 122 = 169, which is 132. ✓
Watch out. OT is the hypotenuse, so it is the one you square and then subtract from, not add to.
- Work out the value of x.
- Work out the length of TA.
Method. Tangents from an external point are equal in length, so the two expressions are equal. Solve the equation, then substitute.
- Use the tangent fact. Tangents drawn from the same point to a circle are equal in length.TA = TB (tangents from an external point are equal in length)
- Write the equation. The two expressions are equal because of the circle theorem, so we can put an equals sign between them.5x − 6 = 3x + 2 (written the other way round so that the x terms stay positive)
- Subtract 3x from both sides. We want all the x terms on one side. Doing the same to both sides keeps the equation balanced.5x − 6 − 3x = 3x + 2 − 3x
2x − 6 = 2 - Add 6 to both sides. We want the number terms on the other side.2x = 2 + 6
2x = 8 - Divide both sides by 2. 2x means 2 × x, so dividing by 2 leaves x on its own.x = 8 ÷ 2 = 4
- Work out the length. Put x back into the expression for TA.TA = 3 × 4 + 2 = 14 cm
Check. TB = 5 × 4 − 6 = 14 cm, the same as TA. ✓
Watch out. Check the final length is positive. A negative length means a mistake.
Method. Use the right angles at the points of contact to apply Pythagoras' theorem to both triangles.
- Draw the line OT. It is the hypotenuse of both triangles OAT and OBT.triangles OAT and OBT share the side OT
- Find the right angles. A tangent is perpendicular to the radius at the point of contact.angle OAT = angle OBT = 90° (the angle between a tangent and a radius is 90°)
- Write Pythagoras for each triangle. In a right-angled triangle, the hypotenuse squared is the sum of the other two sides squared.OT2 = OA2 + TA2 and OT2 = OB2 + TB2
- Use the equal radii. OA and OB are radii of the same circle.OA = OB, so OA2 = OB2
- Compare the two equations. The left sides are the same, and the first term on each right side is the same, so the second terms must be equal.TA2 = OT2 − OA2 = OT2 − OB2 = TB2
- Take the square root. Lengths are positive.TA = TB. ✓
Check. Test: radius 5, OT = 13. Then TA = √(169 − 25) = 12 and TB = 12. ✓
Watch out. A second proof uses congruent triangles: the triangles have a right angle, the same hypotenuse and one equal side (RHS), so they are congruent.
Exercise 26F Tangents and radii
Non-calculatorKey words. A tangent is a straight line that touches a circle at exactly one point, the point of contact.
Method. A tangent is at right angles to the radius at the point where it touches the circle. That makes triangle OAT right-angled.
- Draw the triangle. Join O to A and to T. The radius OA, the tangent TA and the line OT make triangle OAT.triangle OAT
- Find the angle at the point of contact. A tangent touches the circle at one point and is perpendicular to the radius there.Angle OAT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Use the angle sum of triangle OAT. The three angles add up to 180°. Subtract the two we know.Angle AOT = 180° − 90° − 25° = 65° (the angles in a triangle add up to 180°)
Check. 65° + 25° + 90° = 180°. ✓
Watch out. The right angle is at A, the point of contact, between the radius and the tangent.
Key words. A tangent is a straight line that touches a circle at exactly one point, the point of contact.
Method. A tangent is at right angles to the radius at the point where it touches the circle. That makes triangle OAT right-angled.
- Draw the triangle. Join O to A and to T. The radius OA, the tangent TA and the line OT make triangle OAT.triangle OAT
- Find the angle at the point of contact. A tangent touches the circle at one point and is perpendicular to the radius there.Angle OAT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Use the angle sum of triangle OAT. The three angles add up to 180°. Subtract the two we know.Angle AOT = 180° − 90° − 35° = 55° (the angles in a triangle add up to 180°)
Check. 55° + 35° + 90° = 180°. ✓
Watch out. The right angle is at A, the point of contact, between the radius and the tangent.
Method. A tangent is at right angles to the radius at the point where it touches the circle, so triangle OAT has a right angle at A.
- Draw the triangle. Join O to A and to T. The radius OA, the tangent TA and the line OT make triangle OAT.triangle OAT
- Find the angle at the point of contact. A tangent is perpendicular to the radius at the point of contact.Angle OAT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Use the angle sum of triangle OAT. The three angles add up to 180°.Angle ATO = 180° − 90° − 60° = 30° (the angles in a triangle add up to 180°)
Check. 30° + 60° + 90° = 180°. ✓
Watch out. Do not assume the angle at T is 90°. The right angle is at the point of contact.
Method. A tangent is at right angles to the radius at the point where it touches the circle, so triangle OAT has a right angle at A.
- Draw the triangle. Join O to A and to T. The radius OA, the tangent TA and the line OT make triangle OAT.triangle OAT
- Find the angle at the point of contact. A tangent is perpendicular to the radius at the point of contact.Angle OAT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Use the angle sum of triangle OAT. The three angles add up to 180°.Angle ATO = 180° − 90° − 50° = 40° (the angles in a triangle add up to 180°)
Check. 40° + 50° + 90° = 180°. ✓
Watch out. Do not assume the angle at T is 90°. The right angle is at the point of contact.
Method. The two radii and the two tangents make a quadrilateral OATB with two right angles. Use the angle sum 360°.
- Draw the radii OA and OB. They go to the two points of contact and make a quadrilateral OATB with the two tangents.quadrilateral OATB
- Mark the right angles. Each tangent is perpendicular to the radius drawn to its point of contact.Angle OAT = 90° and angle OBT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Use the quadrilateral OATB. Its four angles add up to 360°. Subtract the three we know.Angle AOB = 360° − 90° − 90° − 56° = 124° (the angles in a quadrilateral add up to 360°)
Check. 360° = 360°. ✓ A shortcut for later: angle AOB + angle ATB = 180°.
Watch out. The quadrilateral OATB is a kite. Its two right angles are at A and B, opposite each other.
Method. The two radii and the two tangents make a quadrilateral OATB with two right angles. Use the angle sum 360°.
- Draw the radii OA and OB. They go to the two points of contact and make a quadrilateral OATB with the two tangents.quadrilateral OATB
- Mark the right angles. Each tangent is perpendicular to the radius drawn to its point of contact.Angle OAT = 90° and angle OBT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Use the quadrilateral OATB. Its four angles add up to 360°. Subtract the three we know.Angle AOB = 360° − 90° − 90° − 80° = 100° (the angles in a quadrilateral add up to 360°)
Check. 360° = 360°. ✓ A shortcut for later: angle AOB + angle ATB = 180°.
Watch out. The quadrilateral OATB is a kite. Its two right angles are at A and B, opposite each other.
Method. The two radii and the two tangents make a quadrilateral OATB with two right angles. Use the angle sum 360°.
- Draw the radii OA and OB. They go to the two points of contact and make a quadrilateral OATB with the two tangents.quadrilateral OATB
- Mark the right angles. Each tangent is perpendicular to the radius drawn to its point of contact.Angle OAT = 90° and angle OBT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Use the quadrilateral OATB. Its four angles add up to 360°. Subtract the three we know.Angle AOB = 360° − 90° − 90° − 110° = 70° (the angles in a quadrilateral add up to 360°)
Check. 360° = 360°. ✓ A shortcut for later: angle AOB + angle ATB = 180°.
Watch out. The quadrilateral OATB is a kite. Its two right angles are at A and B, opposite each other.
Method. First find angle AOB from the kite OATB, then use the isosceles triangle made by two radii.
- Find angle AOB. Quadrilateral OATB has right angles at A and B (tangent perpendicular to radius). Its angles add up to 360°.Angle AOB = 360° − 90° − 90° − 72° = 108° (a tangent is perpendicular to the radius at the point of contact; the angles in a quadrilateral add up to 360°)
- Spot the isosceles triangle. OA and OB are radii, so triangle OAB is isosceles and its base angles are equal.Angle OAB = angle OBA (the base angles of an isosceles triangle are equal)
- Use the angle sum in triangle OAB. Subtract the angle at O and share what is left.Angle OAB = (180° − 108°) ÷ 2 = 72° ÷ 2 = 36° (the angles in a triangle add up to 180°)
Check. Angle OAT = 90° and angle OAB + angle BAT should be 90°. Triangle TAB is isosceles with base angles (180° − 72°) ÷ 2 = 54°, and 36° + 54° = 90°. ✓
Watch out. Both triangles OAB and TAB are isosceles. Be careful about which one you are using.
Method. First find angle AOB from the kite OATB, then use the isosceles triangle made by two radii.
- Find angle AOB. Quadrilateral OATB has right angles at A and B (tangent perpendicular to radius). Its angles add up to 360°.Angle AOB = 360° − 90° − 90° − 100° = 80° (a tangent is perpendicular to the radius at the point of contact; the angles in a quadrilateral add up to 360°)
- Spot the isosceles triangle. OA and OB are radii, so triangle OAB is isosceles and its base angles are equal.Angle OAB = angle OBA (the base angles of an isosceles triangle are equal)
- Use the angle sum in triangle OAB. Subtract the angle at O and share what is left.Angle OAB = (180° − 80°) ÷ 2 = 100° ÷ 2 = 50° (the angles in a triangle add up to 180°)
Check. Angle OAT = 90° and angle OAB + angle BAT should be 90°. Triangle TAB is isosceles with base angles (180° − 100°) ÷ 2 = 40°, and 50° + 40° = 90°. ✓
Watch out. Both triangles OAB and TAB are isosceles. Be careful about which one you are using.
Method. Two tangents from one point are equal in length, so triangle TAB is isosceles.
- Show triangle TAB is isosceles. Tangents drawn from the same point to a circle are equal in length.TA = TB (tangents from an external point are equal in length)
- Write the base angles. The base angles of an isosceles triangle are equal: the base angles are at A and B.Angle TAB = angle TBA
- Use the angle sum. Subtract the angle at T and share the rest equally.Angle TAB = (180° − 56°) ÷ 2 = 124° ÷ 2 = 62° (the angles in a triangle add up to 180°)
Check. 62° + 62° + 56° = 180°. ✓
Watch out. Say why the triangle is isosceles: the tangents from T are equal. Do not just say "it looks isosceles".
Method. Two tangents from one point are equal in length, so triangle TAB is isosceles.
- Show triangle TAB is isosceles. Tangents drawn from the same point to a circle are equal in length.TA = TB (tangents from an external point are equal in length)
- Write the base angles. The base angles of an isosceles triangle are equal: the base angles are at A and B.Angle TAB = angle TBA
- Use the angle sum. Subtract the angle at T and share the rest equally.Angle TAB = (180° − 76°) ÷ 2 = 104° ÷ 2 = 52° (the angles in a triangle add up to 180°)
Check. 52° + 52° + 76° = 180°. ✓
Watch out. Say why the triangle is isosceles: the tangents from T are equal. Do not just say "it looks isosceles".
Method. The radius meets the tangent at a right angle, so triangle OAT is right-angled and we can use Pythagoras' theorem.
- Find the right angle. A tangent is perpendicular to the radius at the point of contact.Angle OAT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Write Pythagoras for triangle OAT. The longest side is OT, opposite the right angle.OT2 = OA2 + TA2
132 = 122 + TA2 - Rearrange. Take the known square away from both sides.TA2 = 169 − 144 = 25
- Take the square root. A length is positive.TA = √25 = 5 cm
Check. 122 + 52 = 169, which is 132. ✓
Watch out. OT is the hypotenuse, so it is the one you square and then subtract from, not add to.
Method. The radius meets the tangent at a right angle, so triangle OAT is right-angled and we can use Pythagoras' theorem.
- Find the right angle. A tangent is perpendicular to the radius at the point of contact.Angle OAT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Write Pythagoras for triangle OAT. The longest side is OT, opposite the right angle.OT2 = OA2 + TA2
172 = 82 + TA2 - Rearrange. Take the known square away from both sides.TA2 = 289 − 64 = 225
- Take the square root. A length is positive.TA = √225 = 15 cm
Check. 82 + 152 = 289, which is 172. ✓
Watch out. OT is the hypotenuse, so it is the one you square and then subtract from, not add to.
Method. The radius meets the tangent at a right angle, so triangle OAT is right-angled and we can use Pythagoras' theorem.
- Find the right angle. A tangent is perpendicular to the radius at the point of contact.Angle OAT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Write Pythagoras for triangle OAT. The longest side is OT, opposite the right angle.OT2 = OA2 + TA2
102 = 62 + TA2 - Rearrange. Take the known square away from both sides.TA2 = 100 − 36 = 64
- Take the square root. A length is positive.TA = √64 = 8 cm
Check. 62 + 82 = 100, which is 102. ✓
Watch out. OT is the hypotenuse, so it is the one you square and then subtract from, not add to.
Method. The radius meets the tangent at a right angle, so triangle OAT is right-angled and we can use Pythagoras' theorem.
- Find the right angle. A tangent is perpendicular to the radius at the point of contact.Angle OAT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Write Pythagoras for triangle OAT. The longest side is OT, opposite the right angle.OT2 = OA2 + TA2
252 = 72 + TA2 - Rearrange. Take the known square away from both sides.TA2 = 625 − 49 = 576
- Take the square root. A length is positive.TA = √576 = 24 cm
Check. 72 + 242 = 625, which is 252. ✓
Watch out. OT is the hypotenuse, so it is the one you square and then subtract from, not add to.
Method. Triangle OAT has a right angle at A, so use Pythagoras with OT as the longest side.
- Find the right angle. A tangent is perpendicular to the radius at the point of contact.Angle OAT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Write Pythagoras for triangle OAT. OT is the hypotenuse (opposite the right angle).OT2 = 82 + 152
- Work out the squares and add. OT2 = 64 + 225 = 289
- Take the square root. A distance is positive.OT = √289 = 17 cm
Check. 172 − 82 = 225 = 152. ✓
Watch out. The hypotenuse is added to nothing: add the two shorter sides' squares to get the hypotenuse squared.
Method. The radius meets the tangent at 90°, so triangle OAT is right-angled. Use Pythagoras to find the missing short side, which is the radius.
- Find the right angle. A tangent is perpendicular to the radius at the point of contact.Angle OAT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Write Pythagoras for triangle OAT. OT is the hypotenuse.OT2 = OA2 + TA2
152 = OA2 + 122 - Rearrange for OA2. OA2 = 225 − 144 = 81
- Take the square root. OA = √81 = 9 cm
Check. 92 + 122 = 225 = 152. ✓
Watch out. Subtract to find a shorter side. Add only when the missing side is the hypotenuse.
Method. Tangents from an external point are equal in length, so the two expressions are equal. Solve the equation, then substitute.
- Use the tangent fact. Tangents drawn from the same point to a circle are equal in length.TA = TB (tangents from an external point are equal in length)
- Write the equation. The two expressions are equal because of the circle theorem, so we can put an equals sign between them.3x + 1 = 2x + 8 (written the other way round so that the x terms stay positive)
- Subtract 2x from both sides. We want all the x terms on one side. Doing the same to both sides keeps the equation balanced.3x + 1 − 2x = 2x + 8 − 2x
x + 1 = 8 - Subtract 1 from both sides. We want the number terms on the other side.x = 8 − 1
x = 7 - Read off x. x = 7
- Work out the length. Put x back into the expression for TA.TA = 2 × 7 + 8 = 22 cm
Check. TB = 3 × 7 + 1 = 22 cm, the same as TA. ✓
Watch out. Check the final length is positive. A negative length means a mistake.
Method. Tangents from an external point are equal in length, so the two expressions are equal. Solve the equation, then substitute.
- Use the tangent fact. Tangents drawn from the same point to a circle are equal in length.TA = TB (tangents from an external point are equal in length)
- Write the equation. The two expressions are equal because of the circle theorem, so we can put an equals sign between them.3x + 1 = 2x + 8 (written the other way round so that the x terms stay positive)
- Subtract 2x from both sides. We want all the x terms on one side. Doing the same to both sides keeps the equation balanced.3x + 1 − 2x = 2x + 8 − 2x
x + 1 = 8 - Subtract 1 from both sides. We want the number terms on the other side.x = 8 − 1
x = 7 - Read off x. x = 7
- Work out the length. Put x back into the expression for TA.TA = 2 × 7 + 8 = 22 cm
Check. TB = 3 × 7 + 1 = 22 cm, the same as TA. ✓
Watch out. Check the final length is positive. A negative length means a mistake.
Method. Tangents from an external point are equal in length, so the two expressions are equal. Solve the equation, then substitute.
- Use the tangent fact. Tangents drawn from the same point to a circle are equal in length.TA = TB (tangents from an external point are equal in length)
- Write the equation. The two expressions are equal because of the circle theorem, so we can put an equals sign between them.5x − 2 = 4x + 3 (written the other way round so that the x terms stay positive)
- Subtract 4x from both sides. We want all the x terms on one side. Doing the same to both sides keeps the equation balanced.5x − 2 − 4x = 4x + 3 − 4x
x − 2 = 3 - Add 2 to both sides. We want the number terms on the other side.x = 3 + 2
x = 5 - Read off x. x = 5
- Work out the length. Put x back into the expression for TA.TA = 4 × 5 + 3 = 23 cm
Check. TB = 5 × 5 − 2 = 23 cm, the same as TA. ✓
Watch out. Check the final length is positive. A negative length means a mistake.
Method. Tangents from an external point are equal in length, so the two expressions are equal. Solve the equation, then substitute.
- Use the tangent fact. Tangents drawn from the same point to a circle are equal in length.TA = TB (tangents from an external point are equal in length)
- Write the equation. The two expressions are equal because of the circle theorem, so we can put an equals sign between them.5x − 2 = 4x + 3 (written the other way round so that the x terms stay positive)
- Subtract 4x from both sides. We want all the x terms on one side. Doing the same to both sides keeps the equation balanced.5x − 2 − 4x = 4x + 3 − 4x
x − 2 = 3 - Add 2 to both sides. We want the number terms on the other side.x = 3 + 2
x = 5 - Read off x. x = 5
- Work out the length. Put x back into the expression for TA.TA = 4 × 5 + 3 = 23 cm
Check. TB = 5 × 5 − 2 = 23 cm, the same as TA. ✓
Watch out. Check the final length is positive. A negative length means a mistake.
Method. The quadrilateral OATB has two right angles, so the other two angles add up to 180°. Make an equation and solve it.
- Find the angles at A and B. Each tangent is perpendicular to the radius at its point of contact.Angle OAT = angle OBT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Use the quadrilateral OATB. Its angles add up to 360°. Two of them are 90°, so the other two add up to 360° − 180° = 180°.(4x − 6) + (3x + 46) = 180 (the angles in a quadrilateral add up to 360°)
- Collect like terms. Add the x terms together and the numbers together.7x + 40 = 180
- Subtract 40 from both sides. We want the number terms on the other side.7x = 180 − 40
7x = 140 - Divide both sides by 7. 7x means 7 × x, so dividing by 7 leaves x on its own.x = 140 ÷ 7 = 20
- Work out the angles. Put x back into each expression.Angle ATB = 4 × 20 − 6 = 74°
Angle AOB = 3 × 20 + 46 = 106°
Check. 74° + 106° = 180°. ✓
Watch out. The two tangent right angles use up 180° of the 360°. The equation is "the other two add to 180°".
Method. The quadrilateral OATB has two right angles, so the other two angles add up to 180°. Make an equation and solve it.
- Find the angles at A and B. Each tangent is perpendicular to the radius at its point of contact.Angle OAT = angle OBT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Use the quadrilateral OATB. Its angles add up to 360°. Two of them are 90°, so the other two add up to 360° − 180° = 180°.(4x − 6) + (3x + 46) = 180 (the angles in a quadrilateral add up to 360°)
- Collect like terms. Add the x terms together and the numbers together.7x + 40 = 180
- Subtract 40 from both sides. We want the number terms on the other side.7x = 180 − 40
7x = 140 - Divide both sides by 7. 7x means 7 × x, so dividing by 7 leaves x on its own.x = 140 ÷ 7 = 20
- Work out the angles. Put x back into each expression.Angle ATB = 4 × 20 − 6 = 74°
Angle AOB = 3 × 20 + 46 = 106°
Check. 74° + 106° = 180°. ✓
Watch out. The two tangent right angles use up 180° of the 360°. The equation is "the other two add to 180°".
Method. Use the tangent-radius right angles to find the angle at the centre, then halve it for the angle at the circumference.
- Find the angles at A and B. Each tangent is perpendicular to the radius at its point of contact.Angle OAT = angle OBT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Find the angle at the centre. Quadrilateral OATB has angles that add up to 360°.Angle AOB = 360° − 90° − 90° − 64° = 116° (the angles in a quadrilateral add up to 360°)
- Halve it. Angle ACB is at the circumference and stands on the same arc AB as angle AOB.Angle ACB = 116° ÷ 2 = 58° (the angle at the centre is twice the angle at the circumference)
Check. Second route (alternate segment): triangle TAB is isosceles, so angle TAB = (180° − 64°) ÷ 2 = 58°, and by the alternate segment theorem this equals angle ACB. ✓
Watch out. There are two circle theorems used here. Name each one as you use it.
Method. Use the tangent-radius right angles to find the angle at the centre, then halve it for the angle at the circumference.
- Find the angles at A and B. Each tangent is perpendicular to the radius at its point of contact.Angle OAT = angle OBT = 90° (a tangent is perpendicular to the radius at the point of contact)
- Find the angle at the centre. Quadrilateral OATB has angles that add up to 360°.Angle AOB = 360° − 90° − 90° − 80° = 100° (the angles in a quadrilateral add up to 360°)
- Halve it. Angle ACB is at the circumference and stands on the same arc AB as angle AOB.Angle ACB = 100° ÷ 2 = 50° (the angle at the centre is twice the angle at the circumference)
Check. Second route (alternate segment): triangle TAB is isosceles, so angle TAB = (180° − 80°) ÷ 2 = 50°, and by the alternate segment theorem this equals angle ACB. ✓
Watch out. There are two circle theorems used here. Name each one as you use it.
Method. Split the kite into two equal right-angled triangles, OAT and OBT, and work out the area of each.
- Find the right angles. A tangent is perpendicular to the radius at the point of contact.angle OAT = 90° (the angle between a tangent and a radius is 90°)
- Area of triangle OAT. Its base OA = 5 cm and its height AT = 12 cm are at right angles. Area = ½ × base × height.½ × 5 × 12 = 30 cm2
- Double it. Triangle OBT is identical (tangents from T are equal and the radii are equal).2 × 30 = 60 cm2
Check. The kite has diagonals OT = 13 cm and AB. Using area = ½ × 13 × AB = 60 gives AB = 120 ÷ 13 ≈ 9.2 cm, which is less than the diameter 10 cm. ✓
Watch out. The kite is made of two triangles. Do not use the radius as the height of the whole kite.
Method. Use the right angles at the points of contact and then a congruence condition for right-angled triangles (RHS).
- Find the right angles. The angle between a tangent and a radius is 90°.angle OAT = angle OBT = 90°
- Find a common side. OT is the hypotenuse of both triangles.OT is common
- Find another equal pair. OA and OB are radii of the same circle.OA = OB
- Name the condition. Right angle, hypotenuse and one other side equal: RHS.Triangles OAT and OBT are congruent (RHS).
- Match the angles at T. Congruent triangles have equal matching angles.angle ATO = angle BTO, so OT bisects angle ATB. ✓
Check. Test: with t = 64°, each half is 32°. ✓
Watch out. RHS only works for right-angled triangles. Say “right angle, hypotenuse, side”.
Method. Show triangles OAM and OBM are congruent, so the angles at M are equal, then use the straight line AMB.
- Equal sides. OA and OB are radii, and OM is shared.OA = OB and OM is common
- Equal angles at O. From the previous question, OT bisects angle AOB (the kite is symmetrical about OT).angle AOM = angle BOM
- Congruent triangles. Two sides and the included angle are equal (SAS).Triangles OAM and OBM are congruent.
- Equal angles at M. Matching angles are equal.angle OMA = angle OMB
- Straight line. A, M and B are in a straight line, so these two equal angles add up to 180°.each is 180° ÷ 2 = 90°. So OT is perpendicular to AB. ✓
Check. In a kite the diagonals cross at right angles. ✓
Watch out. This is the same fact as “the line of symmetry of a kite is perpendicular to the other diagonal”.