Circle Theorems — Part 2
Cover tangent-radius, tangents from a point, and the alternate segment theorem in MYP Maths Year 5. Mixed circle theorem problem-solving for Standard level students.
What This Topic Covers
Circle Theorems Part 2 covers three further theorems that students at MYP Year 5 Standard level need to know. These theorems involve tangents and the relationships between angles formed outside and on the circle. This page covers different theorems from Part 1 — no content is repeated.
Theorem 4: Tangent–Radius Relationship
A tangent to a circle is perpendicular to the radius drawn to the point of tangency.
This means the angle between a tangent line and the radius at the point of contact is always 90°. It is often used in combination with Pythagoras' theorem when a triangle is formed between a tangent, a radius, and an external line.
Theorem 5: Tangents from an External Point
Two tangent lines drawn from the same external point to a circle are equal in length.
This creates an isosceles triangle, which students can use to find missing lengths and angles in diagrams involving external tangent lines.
Theorem 6: Alternate Segment Theorem
The angle between a tangent to a circle and a chord drawn from the point of tangency equals the inscribed angle subtending the same chord on the opposite side.
This is one of the more counterintuitive circle theorems. Students need to identify the chord, the tangent, and the relevant inscribed angle carefully. A common approach is to label the angle in question, then locate its alternate-segment counterpart.
Common Mistakes
- Misidentifying which arc the alternate segment angle subtends
- Forgetting that tangent–radius gives a right angle and not calculating accordingly
- Assuming the two tangent lengths are only equal for specific types of circles — this always holds
- Not identifying the tangent line correctly when it extends beyond the point of tangency
MYP Question Style
Harder Criterion A tasks combine theorems from Part 1 and Part 2 in a single diagram. Students may need to use the tangent-radius right angle, an alternate segment angle, and an angle-at-centre relationship in the same problem. Full marks require both correct values and correct theorem citations.
Practice Approach
Isolate each theorem in practice first. Then work through mixed problems where you must decide which theorem applies to each angle. Pay close attention to how tangent lines are drawn in diagrams — the direction they extend can obscure which angle is the alternate segment angle.