☰ Contents · Geometry

A line and a circle. The tangent

Lessons 36 · 1 lessons · A. A. Rahimqoriyev, M. A. To‘xtaxo‘jayeva. Geometry, Grade 8 (textbook for general secondary schools), revised 3rd edition. Yangiyo‘l Poligraf Servis, Tashkent, 2014
36

A line and a circle. The tangent

Textbook: pp. 111–113
GoalKnow the possible positions of a line and a circle and the properties of a tangent.
New words
tangent · urinmapoint of tangency · urinish nuqtasisecant · kesuvchidistance from the centre to the line · markazdan to‘g‘ri chiziqqacha masofa
Explanation

We compare the distance d from the centre O to a line with the radius R. If d > R, the line and the circle have no common point; if d = R, they have exactly one common point; if d < R, they have two common points and the line is called a secant. A line that has exactly one common point with a circle is a tangent, and the common point is the point of tangency. A tangent is perpendicular to the radius drawn to the point of tangency; the converse is also true: a line through the end of a radius perpendicular to it is a tangent. The two tangent segments drawn from a point outside the circle are equal.

Worked examples
R = 9, d = 12: d > R, so the line and the circle have no common point.
From a point A at distance 13 from the centre a tangent is drawn to a circle with R = 5. The tangent segment satisfies 13² − 5² = 144, so it is 12.
Class activity

“Touching ruler”: place a ruler against the edge of a round plate and check with a set square that the angle between the ruler and the radius at the touching point is a right angle.

Practice
1
R = 10 and d = 6. How many common points do the line and the circle have?
2
From a point 17 from the centre a tangent is drawn to a circle with R = 8. How long is the tangent segment?
3
Two tangents are drawn to a circle from an outside point and one segment is 14 cm. What is the sum of both segments (cm)?
4
Why is a tangent perpendicular to the radius at the point of tangency?