We will organize the relationships between circles and lines and learn how to construct a tangent to a circle. Tangents are studied in greater detail in Grade 9, but let us first learn their basic properties and construction method in Grade 7.
Positions of a Circle and a Line
There are three possible relationships between a circle and a line.
- Intersect at two points: The line and circle have two common points. The line cuts off a chord.
- Are tangent: The line and circle intersect at only one point. This line is called a tangent, and the common point is called the point of tangency.
- Do not intersect: They have no common points.
The relationship is determined by the distance d from the center of the circle to the line and the radius r of the circle.
d ＜ r → Intersect at two points
d ＝ r → Are tangent
d ＞ r → Do not intersect
Important Property of Tangents
For a tangent to circle O, the following property holds.
A tangent is perpendicular to the radius passing through the point of tangency. In other words, at point of tangency T, tangent ⊥ OT.
A circle O has radius 5 cm. State the relationship between the circle and line ℓ when the distance between center O and line ℓ is as follows.
(1) d = 3 cm (2) d = 5 cm (3) d = 7 cm
Solution:
(1) Since d ＜ r, they intersect at two points.
(2) Since d ＝ r, they are tangent (one common point).
(3) Since d ＞ r, they do not intersect.
Constructing a Tangent (with T as the Point of Tangency)
Procedure for constructing a line tangent to circle O that passes through point T on the circle:
① Draw radius OT.
② Draw a line through T perpendicular to OT (construct the perpendicular) → This is the tangent.
Use the property that a tangent is perpendicular to the radius passing through the point of tangency.
Construct the tangent to circle O with point P on the circle as the point of tangency.
Solution:
① Draw radius OP.
② Construct a line through P perpendicular to OP (construct the perpendicular through point P on line OP).
③ The resulting line is tangent to circle O.
- Position of a circle and line: intersect (d<r) · tangent (d=r) · separate (d>r)
- A tangent is perpendicular to the radius passing through the point of tangency.
- Constructing a tangent: Draw radius OT, then draw a perpendicular through T.
- Point of tangency: the only common point of a circle and its tangent
Practice Problems
- Circle O has radius 6 cm, and the distance from center O to line ℓ is 6 cm. State the relationship between the circle and line.
- State the procedure for constructing a tangent whose point of tangency is point A on circle O.
- What is the angle formed by a tangent and the radius passing through the point of tangency?
Answers and Explanations
- Answer: They are tangent.
Explanation: Since d = r = 6, the circle and line touch at one point. - Answer: Draw radius OA, then construct a line through A perpendicular to OA (construct the perpendicular through point A on the line).
Explanation: A tangent is perpendicular to the radius passing through the point of tangency. - Answer: 90°
Explanation: The tangent is perpendicular to the radius passing through the point of tangency.