4 high-yield, syllabus-aligned questions on Circle geometry, covering 10 individually creditable mark points. Each one shows the answer that scores full marks under our guide, which phrase earns which mark, and a common incomplete answer — so you can see the difference rather than guess at it.
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Every question and solution is reviewed by examiners versed in WAEC and JAMB, each with more than thirty years of experience.
Mark allocations are TopMarks’ own, written against the published syllabus and Chief Examiners’ reports. They are a guide to how these answers are usually credited, not an official marking scheme.
State3 marks
State three circle theorems relating to angles.
TopMarks model answer
The angle which an arc subtends at the centre of a circle is twice the angle it subtends at any point on the remaining part of the circumference. Angles in the same segment of a circle are equal. The angle in a semicircle is a right angle. In addition, the opposite angles of a cyclic quadrilateral are supplementary, that is they add up to 180°.
Suggested marking guide
1Angle at centre is twice the angle at the circumference
2Angles in the same segment are equal
3Angle in a semicircle, or cyclic quadrilateral
Why it scores
Quote theorems in full. “The angle at the centre is twice the angle at the circumference” earns the mark; “angle at centre theorem” does not, because the examiner cannot tell whether you know what it says. In a proof, quoting the theorem by its full statement is also worth a reason mark on every line.
A common incomplete answer
“Angles in the same segment of a circle are equal.”
What it costs: Two marks of three. One theorem is stated correctly and credited, but the question asked for three.
Key ideas to include: angle at the centre, same segment, semicircle, cyclic quadrilateral, supplementary.
Calculate2 marks
In a circle with centre O, angle AOB = 96°. C is a point on the major arc. Calculate angle ACB.
TopMarks model answer
The angle which arc AB subtends at the centre is twice the angle it subtends at the circumference. So angle AOB = 2 x angle ACB. Therefore angle ACB = 96° ÷ 2 = 48°.
Suggested marking guide
1The correct theorem quoted
2Correct answer
The working, line by line
1Angle at centre = 2 x angle at circumference (same arc AB) — Quote the theorem. In circle geometry the REASON is a mark, separate from the number.
296° = 2 x angle ACB — Set up the relationship before you divide.
3angle ACB = 48° — Include the degree sign.
Why it scores
In circle geometry the reason is worth as much as the answer. Every line of a solution should read “angle X = … (reason)”. A candidate who writes 48° with no justification has shown the examiner a number, not a piece of geometry.
A common incomplete answer
“Angle ACB = 48°”
What it costs: One mark of two. The answer is correct, but no theorem is quoted, so the reason mark is lost.
Key ideas to include: angle at the centre, angle at the circumference, subtended by the same arc, major arc.
Find3 marks
PQRS is a cyclic quadrilateral in which angle P = 3x and angle R = x + 40. Find x.
TopMarks model answer
P and R are opposite angles of a cyclic quadrilateral, and the opposite angles of a cyclic quadrilateral are supplementary, so angle P + angle R = 180°. Therefore 3x + (x + 40) = 180. So 4x + 40 = 180, giving 4x = 140 and x = 35.
Suggested marking guide
1The supplementary-angles theorem quoted
2Equation formed and simplified
3Correct value of x
The working, line by line
1P and R are OPPOSITE angles, so they are supplementary — Check first that the two angles named really are opposite. Adjacent angles of a cyclic quadrilateral are not supplementary.
23x + (x + 40) = 180 — Form the equation from the theorem.
34x + 40 = 180 → 4x = 140 → x = 35 — Collect and solve.
4Check: P = 105°, R = 75°, and 105 + 75 = 180 ✓ — Substituting back confirms the answer in seconds.
Why it scores
Three marks and three distinct things to write: the theorem, the equation, the solution. Candidates lose the first because it feels obvious once the equation is written — but the examiner marks what is on the page, not what was in the candidate’s head.
A common incomplete answer
“3x + x + 40 = 180”
What it costs: Two marks of three. The equation is right, and “180” just carries the theorem mark, but the equation is never solved, so x is never found.
Key ideas to include: cyclic quadrilateral, opposite angles, supplementary, form an equation.
Explain2 marks
Explain why a tangent to a circle is perpendicular to the radius drawn to the point of contact, and state how this fact is used in solving problems.
TopMarks model answer
A tangent touches the circle at exactly one point, and the radius drawn to that point of contact is the shortest distance from the centre to the tangent line; the shortest distance from a point to a line is always along the perpendicular to that line. In problems, this means that the triangle formed by the centre, the point of contact and an external point is right-angled at the point of contact, so Pythagoras’ theorem and the trigonometric ratios can be applied to it.
Suggested marking guide
1The radius is the shortest distance, hence perpendicular
2It creates a right-angled triangle that can be solved
Why it scores
The second mark is the practical one. The tangent–radius theorem is almost never the answer by itself; it is the step that MANUFACTURES a right-angled triangle, which is what lets you use Pythagoras or trigonometry. Whenever a tangent appears in a question, draw that radius first.
A common incomplete answer
“Because the angle between the tangent and the radius is 90°.”
What it costs: Both marks. This restates the theorem rather than explaining it, and it says nothing about how the fact is used.
Key ideas to include: tangent, point of contact, radius, shortest distance, right-angled triangle.
Practise Circle geometry on real questions
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