Introduction to Circle Theorems
Circle theorems are fundamental properties that relate angles and lines in and around circles. They allow us to find unknown angles by applying established geometric facts. Two important theorems for this objective are: the angle in a semicircle is a right angle (90°), and the angle between a tangent and a radius at the point of contact is also 90°. Understanding these theorems helps solve many problems involving circles and tangents.
- Circle theorems link angles and arcs in circles.
- They provide reliable methods to calculate unknown angles.
- Focus here is on the angle in a semicircle and tangent-radius angle theorems.
The Angle in a Semicircle Theorem
This theorem states that if a triangle is drawn with its hypotenuse as the diameter of a circle, then the angle opposite the diameter is always 90°. This means any point on the circle forming a triangle with the diameter creates a right angle at that point.
To apply this theorem, identify the diameter of the circle and the triangle formed. The angle opposite the diameter is then known to be 90°, which can help find other unknown angles in the figure.
- Diameter forms the hypotenuse of a triangle.
- Angle opposite the diameter on the circle is 90°.
- Use this to find unknown angles in circle problems.
The Angle Between a Tangent and a Radius
A tangent to a circle is a straight line that touches the circle at exactly one point. The radius drawn to the point of contact is always perpendicular to the tangent line. Therefore, the angle between the radius and the tangent line is 90°.
This property is useful when a tangent line and radius are shown in a diagram. Knowing this right angle allows you to calculate other angles in the figure using angle rules such as the sum of angles in a triangle or on a straight line.
- Tangent touches the circle at one point only.
- Radius to the point of contact is perpendicular to the tangent.
- Angle between radius and tangent is 90°.
Applying the Theorems to Find Unknown Angles
When solving problems, first identify if the angle in question relates to a semicircle or involves a tangent and radius. Use the relevant theorem to assign a 90° angle where appropriate.
Next, use other angle facts such as angles on a straight line sum to 180°, angles in a triangle sum to 180°, or vertically opposite angles are equal to find the unknown angles.
Always justify your answer by stating which theorem or property you used and why it applies in the given diagram.
- Identify semicircle or tangent-radius situations.
- Assign 90° angles based on the theorems.
- Use angle sum properties to find other unknown angles.
- Explain reasoning clearly referencing the theorems.
Worked Example: Finding an Unknown Angle Using the Semicircle Theorem
Consider a circle with diameter AB. Point C lies on the circle forming triangle ABC. Given that AB is the diameter, find angle ACB.
Since AB is the diameter, by the angle in a semicircle theorem, angle ACB is 90°. This is because the angle opposite the diameter in a triangle on the circle is always a right angle.
Therefore, angle ACB = 90°. This is the final answer, justified by the semicircle theorem.
- Identify AB as diameter.
- Recognise triangle ABC with C on the circle.
- Apply angle in semicircle theorem: angle ACB = 90°.
Worked Example: Using the Tangent-Radius Theorem
A circle has a tangent line touching it at point T. The radius OT is drawn to the point of contact. Find the angle between the tangent and the radius OT.
By the tangent-radius theorem, the angle between the tangent and radius OT is 90°. This is because the radius is perpendicular to the tangent at the point of contact.
Hence, the angle between the tangent and radius OT is 90°, justified by the tangent-radius theorem.
- Identify tangent line and radius OT.
- Apply tangent-radius theorem: angle between them is 90°.
- State conclusion with justification.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Identify relevant circle elements | Look for diameter, tangent, radius, and points on the circle | Diameter AB with point C on circle forming triangle ABC | Shows which theorem to apply |
| Apply theorem | Assign 90° angle based on semicircle or tangent-radius theorem | Angle ACB = 90° because AB is diameter | Marks awarded for correct theorem application |
| Use angle rules | Sum of angles in triangle or on straight line to find unknowns | Calculate other angles using 180° sum | Demonstrates understanding of angle properties |
| Explain reasoning | State which theorem and why it applies | 'Angle ACB is 90° as it is in a semicircle' | Justification earns method marks |
Active recall
Quick check
1. What is the size of the angle in a semicircle?
90 degrees, because the angle opposite the diameter in a circle is a right angle.
2. What angle is formed between a tangent and the radius at the point of contact?
90 degrees, since the radius is perpendicular to the tangent at the point of contact.
3. How do you justify an angle found using the tangent-radius theorem?
By stating that the radius is perpendicular to the tangent line at the point of contact, so the angle between them is 90°.
4. If AB is the diameter of a circle and C is a point on the circle, what is angle ACB?
Angle ACB is 90°, because it is the angle in a semicircle.