Circle Theorems & Circle Equations

What is the equation of a circle with centre at the origin and radius 6?

• Ax squared + y squared = 36✓
• Bx squared - y squared = 36
• Cx squared + y squared = 6
• Dx squared + y squared = 12

Answer: A circle centred at the origin with radius r has equation x squared + y squared = r squared, because every point on the circle is a distance r from the origin by Pythagoras. With radius 6, this gives x squared + y squared = 36, not 6 or 12, which would come from forgetting to square the radius. Changing the sign to x squared - y squared = 36 instead describes a hyperbola, not a circle.

A circle has equation x squared + y squared = 25. What is its radius?

• Aradius = 5✓
• Bradius = 25
• Cr = 10
• Dradius = 12.5

Answer: Comparing x squared + y squared = 25 with the general form x squared + y squared = r squared shows that r squared = 25, so the radius r is the square root of 25, which is 5. The value 25 is the square of the radius rather than the radius itself, and 12.5 and 10 both come from mishandling the square root. Only 5 correctly undoes the squaring.

A circle has equation x squared + y squared = 25. What is the gradient of the tangent to the circle at the point (3, 4)?

• A-3/4✓
• B4/3
• C-4/3
• D3/4

Answer: The radius drawn from the origin to (3, 4) has gradient 4/3, since it rises 4 and runs 3 from the centre. A tangent to a circle is always perpendicular to the radius at the point of contact, so its gradient is the negative reciprocal of 4/3, which is -3/4. The other options either keep the wrong sign or fail to flip the fraction, so they do not represent a perpendicular line.

Points A, B and C lie on a circle with centre O. The angle at the centre AOC is 100 degrees. What is the angle at the circumference ABC, subtended by the same arc AC?

• A100 degrees
• B40 degrees
• C50 degrees✓
• D200 degrees

Answer: The angle at the centre of a circle is always twice the angle at the circumference when both angles stand on the same arc. Since the angle at the centre AOC is 100 degrees, the angle at the circumference ABC must be half of this, which is 50 degrees. Answering 100 degrees ignores the doubling relationship entirely, and 200 degrees and 40 degrees do not come from correctly halving 100.

An angle at the circumference of a circle, standing on the same arc as the angle at the centre, measures 35 degrees. What is the size of the angle at the centre?

• A17.5 degrees
• B105 degrees
• C35 degrees
• D70 degrees✓

Answer: Since the angle at the centre is twice the angle at the circumference on the same arc, the angle at the centre must be double 35 degrees, giving 70 degrees. Stating 35 degrees mistakenly treats the two angles as equal, and 17.5 degrees applies the halving the wrong way round. There is no theorem that produces 105 degrees from this information.

In a circle with centre O, the reflex angle AOC is 250 degrees, and point B lies on the minor arc AC. What is the size of angle ABC?

• A110 degrees
• B55 degrees
• C125 degrees✓
• D205 degrees

Answer: The angle-at-the-centre theorem still applies when the centre angle is reflex, so the angle at the circumference equals half of the reflex angle rather than half of the remaining non-reflex angle. Half of 250 degrees is 125 degrees. Using the non-reflex angle of 110 degrees directly, or halving it to 55 degrees, applies the theorem to the wrong angle, and 205 degrees does not come from halving either angle correctly.

AB is a diameter of a circle, and C is a point on the circumference. What is the size of angle ACB?

• A90 degrees✓
• B45 degrees
• C180 degrees
• D100 degrees

Answer: Any angle drawn at the circumference from the two ends of a diameter is always a right angle, known as the angle in a semicircle. This means angle ACB must equal 90 degrees regardless of exactly where C sits on the circumference. A value of 180 degrees would make A, C and B collinear rather than forming a triangle, and 45 degrees or 100 degrees do not satisfy this theorem.

A triangle is drawn inside a circle with one side as the diameter. One of the other two angles in the triangle is 35 degrees. What is the size of the third angle?

• A125 degrees
• B35 degrees
• C55 degrees✓
• D145 degrees

Answer: Because one side of the triangle is a diameter, the angle opposite that side, at the point on the circumference, is 90 degrees by the angle-in-a-semicircle theorem. The three angles of any triangle sum to 180 degrees, so the third angle is 180 minus 90 minus 35, which is 55 degrees. Answering 35 degrees repeats the given angle rather than finding the missing one, and 125 degrees or 145 degrees do not make the three angles sum correctly.

Points A, B, C and D lie on a circle. Angle ADB and angle ACB are both subtended by the same arc AB, from the same segment. Angle ADB = 42 degrees. What is the size of angle ACB?

• A42 degrees✓
• B21 degrees
• C48 degrees
• D84 degrees

Answer: Angles subtended by the same arc, from points in the same segment of the circle, are always equal to one another. Since angle ADB and angle ACB both stand on arc AB from the same side, angle ACB must also equal 42 degrees. Halving or doubling the given angle, as 21 degrees and 84 degrees do, or shifting it to 48 degrees, does not follow from this theorem.

Chord XY of a circle subtends angle XPY = 58 degrees at point P, and angle XQY at point Q, with both P and Q lying on the same arc. What is the size of angle XQY?

• A58 degrees✓
• B116 degrees
• C29 degrees
• D148 degrees

Answer: Because P and Q lie on the same arc and both angles are subtended by the same chord XY, the angles-in-the-same-segment theorem states that angle XQY must equal angle XPY exactly. Since angle XPY = 58 degrees, angle XQY is also 58 degrees. The value 29 degrees halves the angle incorrectly, 116 degrees doubles it, and 148 degrees does not relate to 58 degrees by any rule from this theorem.

ABCD is a cyclic quadrilateral. Angle A = 105 degrees. What is the size of the opposite angle C?

• A255 degrees
• B75 degrees✓
• C105 degrees
• D85 degrees

Answer: In any cyclic quadrilateral, opposite angles add up to 180 degrees because they are subtended by the same arc from opposite sides of the circle. Since angle A = 105 degrees, angle C must be 180 minus 105, which is 75 degrees. Stating 105 degrees treats the opposite angles as equal rather than supplementary, and 255 degrees and 85 degrees do not make the pair sum to 180.

In cyclic quadrilateral PQRS, angle P = 80 degrees and angle Q = 95 degrees. What is the size of angle S, the angle opposite Q?

• A175 degrees
• B95 degrees
• C85 degrees✓
• D100 degrees

Answer: Opposite angles in a cyclic quadrilateral sum to 180 degrees, and since S is opposite Q, angle S must equal 180 minus angle Q, which is 180 minus 95, giving 85 degrees. The value 100 degrees is actually the angle opposite P rather than Q, and 175 degrees comes from wrongly adding the two given angles together instead of using the 180-degree rule. Repeating 95 degrees would wrongly treat the opposite angles as equal.

In cyclic quadrilateral WXYZ, angle W is three times the size of the opposite angle Y. What is the size of angle W?

• A45 degrees
• B120 degrees
• C90 degrees
• D135 degrees✓

Answer: Since W and Y are opposite angles in a cyclic quadrilateral, they must sum to 180 degrees, and the question states that W is three times Y. Writing this as 3Y + Y = 180 gives 4Y = 180, so Y = 45 degrees, and W, being three times as large, is 135 degrees. The value 45 degrees is actually angle Y rather than W, and 90 degrees and 120 degrees do not satisfy both the 180-degree sum and the three-times relationship together.

A tangent touches a circle at point A, and O is the centre. What is the angle between the tangent and the radius OA?

• A180 degrees
• B90 degrees✓
• C150 degrees
• D135 degrees

Answer: A tangent to a circle always meets the radius drawn to the point of contact at a right angle. This means the angle between the tangent at A and the radius OA is exactly 90 degrees, regardless of where A lies on the circle. A value of 180 degrees would place the tangent and radius on the same straight line, and 135 degrees or 150 degrees do not satisfy this perpendicularity rule.

A tangent to a circle at point T meets the radius OT, and a chord TS is drawn from T making an angle of 35 degrees with the tangent. Given that the radius is perpendicular to the tangent, what is the size of angle OTS?

• A55 degrees✓
• B35 degrees
• C145 degrees
• D125 degrees

Answer: Because the radius OT is perpendicular to the tangent at T, the full angle between the radius and the tangent is 90 degrees. The chord TS splits this right angle into two parts, one of which is the given 35 degrees between the tangent and the chord, so angle OTS, the remaining part, is 90 minus 35, which is 55 degrees. Repeating 35 degrees ignores this split, and 145 degrees or 125 degrees each apply the wrong operation to the two given values.

Two tangents are drawn from an external point P to touch a circle at points A and B. If PA = 8 cm, what is the length of PB?

• A8 cm✓
• B16 cm
• C6 cm
• D4 cm

Answer: Two tangents drawn to a circle from the same external point are always equal in length, because the two right-angled triangles formed with the centre and each point of contact are congruent. Since PA = 8 cm, the second tangent PB must also equal 8 cm. Halving to 4 cm, doubling to 16 cm, or choosing an unrelated value such as 6 cm does not follow from this equal-tangents property.

From an external point Q, two tangents QC and QD touch a circle at C and D, forming triangle QCD with QC = QD. Angle CQD = 40 degrees. What is the size of angle QCD?

• A70 degrees✓
• B40 degrees
• C50 degrees
• D110 degrees

Answer: Since tangents from the same external point are equal in length, QC = QD, which makes triangle QCD isosceles with the two base angles at C and D equal to one another. The three angles sum to 180 degrees, so the two equal base angles together make 180 minus 40, which is 140 degrees, and each one individually is 70 degrees. Repeating the apex angle of 40 degrees, or not halving the remaining 140 degrees correctly as 110 degrees or 50 degrees do, gives the wrong base angle.

A tangent touches a circle at point T, and a chord TS is drawn from T. The angle between the tangent and the chord is 48 degrees. By the alternate segment theorem, what is the size of the angle in the alternate segment subtended by TS?

• A132 degrees
• B96 degrees
• C142 degrees
• D48 degrees✓

Answer: The alternate segment theorem states that the angle between a tangent and a chord at the point of contact equals the angle in the alternate segment, subtended by that same chord. Since the tangent-chord angle here is 48 degrees, the angle in the alternate segment must also be 48 degrees. Doubling to 96 degrees, or shifting to 142 degrees or the supplement 132 degrees, do not match this direct equality.

A tangent meets a circle at point A, and chord AB makes an angle of 70 degrees with the tangent on one side. Point C lies on the major arc, in the alternate segment. What is the size of angle ACB?

• A110 degrees
• B140 degrees
• C70 degrees✓
• D35 degrees

Answer: By the alternate segment theorem, the angle between the tangent and the chord AB equals the angle subtended by AB at any point in the alternate segment, so angle ACB must equal the given tangent-chord angle of 70 degrees. Halving this to 35 degrees or doubling it to 140 degrees does not reflect the direct equality the theorem describes, and 110 degrees would instead be the supplement of 70 degrees, which applies to a different pair of angles.

AB is a diameter of a circle with centre O, and C is a point on the circumference. Angle BAC = 25 degrees. Using the fact that the angle in a semicircle is 90 degrees, what is the size of angle ABC?

• A155 degrees
• B25 degrees
• C65 degrees✓
• D115 degrees

Answer: Because AB is a diameter, the angle in a semicircle theorem gives angle ACB = 90 degrees. The three angles of triangle ABC sum to 180 degrees, so angle ABC is 180 minus 90 minus 25, which is 65 degrees. Repeating the given 25 degrees answers a different angle in the triangle, and 115 degrees or 155 degrees each come from taking a supplement instead of the correct angle sum.

In a circle with centre O, points A, B, C and D lie on the circumference in order, forming cyclic quadrilateral ABCD. The reflex angle at the centre AOC, measured through D, is 220 degrees, so by the angle-at-the-centre theorem the angle ABC at the circumference is 110 degrees. Using the cyclic quadrilateral theorem, what is the size of the opposite angle ADC?

• A110 degrees
• B250 degrees
• C70 degrees✓
• D140 degrees

Answer: Opposite angles in a cyclic quadrilateral sum to 180 degrees, and since angle ADC is opposite the already-found angle ABC of 110 degrees, angle ADC must equal 180 minus 110, which is 70 degrees. This combines two theorems in sequence: the angle-at-the-centre theorem first gives 110 degrees, and the cyclic quadrilateral theorem then gives the final 70 degrees. Repeating 110 degrees treats the opposite angles as equal rather than supplementary, and 140 degrees and 250 degrees do not correctly complete the pair to 180 degrees.