Semicircle angles and tangent radii
Use the two Core circle theorems with their required conditions and geometric reasons.
Section Paper 4 · Extended · Calculator
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Identify a circle's diameter, radius and tangent.
- Use triangle angle sums.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which chord is a diameter?
What you will learn
- Identify the right angle subtended by a diameter.
- Use the perpendicular radius at a tangent's contact point.
- Combine these facts with triangle angle sums.
A diameter makes a right angle at the circle
If A and B are the endpoints of a diameter and C is another point on the circumference, angle ACB is 90°. This is the angle in a semicircle theorem. The angle's vertex is on the circumference and the opposite side AB must pass through the centre; a random chord does not guarantee a right angle.
If angle CAB is 32°, the remaining angle ABC is 180 − 90 − 32 = 58°. Write both reasons: angle ACB is a right angle because AB is a diameter, and the third angle follows from the triangle total. The triangle can be rotated or drawn with C below AB without changing these conditions.
The radius is perpendicular at contact
A tangent touches the circle at T. The radius OT from centre O to that precise contact point meets the tangent at 90°. A line from the centre to a different circumference point is not guaranteed perpendicular to the tangent. Add the correct radius to a diagram before using the theorem.
For an external point P on the tangent, triangle OTP has a right angle at T. If angle OPT is 38°, angle POT is 52° by the triangle sum. These are the two circle-angle rules required by Core. Extended adds further theorems in the next lesson; do not use a theorem name without identifying the chord, arc or contact point it refers to.
Pause and explain
A tangent touches at T. Which line is perpendicular to it?
Worked example
AB is a diameter and C lies on the circle. Angle CAB = 32°. Find ACB and ABC with reasons.
Show the worked solution
- ACB = 90° because the angle subtended by diameter AB at the circumference is a right angle.
- Angles in triangle ABC total 180°.
- ABC = 180 − 90 − 32 = 58°.
Answer ACB = 90°; ABC = 58°.
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Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
AB is a diameter, C is on the circle and angle ABC = 41°. Find ACB and BAC.
Give me a hint
Use the diameter condition before subtracting from 180°.
Compare my reasoning
- ACB is 90° by the angle in a semicircle theorem.
- BAC = 180 − 90 − 41.
- Thus BAC = 49°; the triangle total checks.
ACB = 90°; BAC = 49°.
Look for these in your work
- Identified the angle opposite the diameter.
- Gave a triangle-sum reason for the second result.
PT is tangent at T to a circle with centre O. Angle OPT = 38°. Find OTP and POT.
Give me a hint
The radius OT meets the tangent PT at contact.
Compare my reasoning
- OTP = 90° because a radius is perpendicular to the tangent at contact.
- POT = 180 − 90 − 38.
- The central angle in this triangle is therefore 52°.
OTP = 90°; POT = 52°.
Look for these in your work
- Used the correct contact radius.
- Distinguished the triangle's vertices.
A student claims that angle ACB is 90° because A, B and C are on a circle. What extra condition is needed, and why?
Give me a hint
Not every inscribed triangle is right-angled.
Compare my reasoning
- The chord AB opposite angle C must be a diameter.
- That requires AB to pass through the circle's centre.
- Without this condition, the semicircle theorem cannot establish the right angle.
AB must be a diameter; three circumference points alone are insufficient.
Look for these in your work
- Checked the theorem's condition.
- Explained why appearance or circle membership alone is insufficient.
Common mistakes
- Treating every chord as a diameter.
- Placing the right angle at the centre for the semicircle theorem.
- Using a radius to the wrong contact point.
What can you explain now?
Is a tangent perpendicular to the radius at its contact point?
Compare with the explanation
Yes, the angle is 90°.
The radius must end at the tangent's exact point of contact; that is the condition of the theorem.
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Make it stick
Tomorrow, sketch both Core circle theorems. Mark the conditions before writing 90° or finding any remaining angle.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.