How do you name the parts of a circle without mixing them up?
You lose marks in circle questions surprisingly fast if you call the wrong thing a chord or a tangent. A lot of the topic is really about spotting what kind of line or region you are looking at, then using the right fact.
A circumference is the outside curved edge of your circle.
The center is the single point inside the circle that sits the same distance from every point on the circumference.
A radius is a segment joining the center to the circumference.
A diameter is a chord that goes through the center, and its length is twice the radius.
A chord is a segment with both endpoints on the circumference.
An arc is just one section of the circumference between two chosen points.
A segment is the area trapped between a chord and the matching arc.
A secant is a full line cutting through the circle in two places.
A tangent is a line that meets the circle once only, at the point of contact.
A common trap is forgetting the difference between a line and a line segment. Your chord has two endpoints and stops there. Your secant keeps going forever in both directions, so even if it contains a chord-shaped part, it is not itself a chord.
If you are staring at a diagram, think of a chord like a phone charging cable stretched between two points on the rim, while a secant is more like a laser line passing straight through and continuing beyond the circle.
Another phrase you need for theorem questions is subtended angle. When two lines join a point to two points on the circle, they make an angle standing on the same chord or arc.
An angle subtended by a chord or arc is the angle formed when you connect a point to the two endpoints of that chord or arc. You can get one at the center or one on the circumference.
Why does a tangent always make a right angle with the radius?
This is one of those facts examiners expect you to spot instantly. If a line touches the circle at one point and you draw the radius to that touching point, the angle there is always a right angle.
The theorem is written as
Here, is the center and is the point where the tangent touches the circle.
Method
- Find the point where the tangent touches the circle.
- Draw or identify the radius from the center to that point.
- Mark the angle between the radius and the tangent as .
Worked example
A line touches a circle at , and is the center. You draw radius . What angle can you state immediately?
- The line is a tangent at .
- is a radius to the point of contact.
- Radius and tangent are perpendicular at the contact point.
- So the angle between and the tangent is .
You can picture it like a bike wheel. The radius is like a spoke from the middle to the tyre, and the tangent is the direction the tyre would move at that exact point. Those two directions meet at right angles.
Do not label a line as a tangent just because the sketch looks like it barely touches the circle. In a proof, you need a reason such as “it meets the radius at ” or “it intersects the circle at exactly one point.”
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