A secant that has shrunk
Slide a secant outward: its chord shrinks to nothing and the two points become one.
The chord and the ruler
A secant PQ cuts a circle at two points A and B. The piece AB inside the circle is a chord.
Now slide the ruler outward, keeping it parallel to PQ.
As the line slides away from the centre, what happens to the chord AB?
What this lesson covers
The idea
A tangent to a circle is a special case of a secant, when the two end points of its corresponding chord coincide.
The chord and the ruler
A secant PQ cuts a circle at two points A and B. The piece AB inside the circle is a chord.
Now slide the ruler outward, keeping it parallel to PQ.
As the line slides away from the centre, what happens to the chord AB?
- It gets shorter
- It gets longer
- It stays the same
Slide the line out
The faint dashed line is the secant PQ. Drag the knob to slide a parallel line outward, on either side of PQ. Watch the chord AB in centimetres.
The two points meet
A tangent to a circle is a special case of a secant, when the two end points of its corresponding chord coincide.
The chord got shorter as the line moved out. When A and B met, the chord was 0 cm and the line touched the circle at one point. On each side of the secant there is one such tangent, so there are exactly two tangents parallel to a given secant, and never more.
Notes
A tangent is a special secant: slide a secant outward until the two end points of its chord coincide. Two tangents are parallel to a given secant, one on each side.
Check yourself
A circle has radius 5 cm. A line cuts it at a distance of 3 cm from the centre. How long is the chord (in cm)?
Answer: 8
Half the chord = √(5² − 3²) = √16 = 4 cm. So the chord is 2 × 4 = 8 cm.
A line slides outward and the chord AB shrinks to 0 cm. What is the line now?
How many tangents can be drawn parallel to a given secant of a circle?
Answer: 2
One tangent on each side of the secant, so 2.
A circle has radius 5 cm and a line is 6 cm from its centre. How does the line meet the circle?
- A secant. A secant has two different common points, so its chord is not 0.
- A tangent — correct. Yes! The two end points of the chord have met, so the line touches the circle at one point.
- A non-intersecting line. The line still touches the circle at the point where A and B met.
- It is a secant.. A secant is closer to the centre than the radius. Here the line is farther than the radius.
- It is a tangent.. A tangent is exactly 5 cm from the centre, the radius. This line is farther away.
- It does not meet the circle. — correct. Yes! The line is farther from the centre than the radius, so it misses the circle.