section7

CIRCLES: DEFINITION OF A CIRCLE, CHORDS, TANGENT AND SECANT LINES.

This section introduces the properties of the circles. We start with the definition of this geometric object. A circle is the set of all points in a plane at a fixed distance from a fixed point in the plane. The fixed point is called the center of the circle. The fixed distance is called the radius of the circle. A segment from a point of the circle to the center is also called a radius. Usually a circle is named by its center e.g. circle O. Sometimes we are also going to specify the radius r of the circle and name the circle as (O, r).

We say that two circles are congruent if they have the same radius.
If two or more circles share the same center then they are called concentric circles.

A segment that connects two distinct points on a circle is called a chord.
A chord that passes through the center of a circle is called a diameter.
A line that intersects with the circle at exactly two points is called a secant (line).

Theorem 7.1 A diameter passing through the middle point of a chord is perpendicular to this chord.

Proof Consider an arbitrary circle O with a chord AB and diameter CD. Let E be the middle point of AB.

We need to show that DC is perpendicular to AB. Indeed, the triangle AOB is an isosceles triangle since AO = OB as radii (plural of radius) of the same circle O. Therefore the median OE is an altitude and OE is perpendicular to AB. So, DC is perpendicular to AB as well.

We would like to discuss one more type of lines that are connected with circles. A line is called tangent to a circle O if this line passes through an arbitrary point X on the circle O and is perpendicular to the radius OX. The point X where the tangent line touches the circle O is called the point of tangency.

Theorem 7.2 If a line t is tangent to a circle O with the tangency point X then X is the only point that t and circle O have in common.

Proof In order to prove this theorem we are going to use the technique that we have discussed in the previous section, namely a proof by contradiction.

Suppose that t and O have two points in common, points X and Y. By definition of a tangent line, OX is perpendicular to t and OY is perpendicular to t.

Then the triangle OXY has two right angles and its third angle should be equal to zero degrees, which is impossible. Therefore our assumption that t and O have two points in common was false and X is the only common point.

EXERCISE SET #7

Problem 1. Suppose that we have two distinct circles (O, r) and (P, q). If r = 25 inches, q = 50 inches and the length of the segment OP is 60 inches, is it possible that circles (O, r) and (P, q) are tangent (i.e. they have exactly one point in common)? Why?

Problem 2. From a given point A on a circle O a diameter AB and a chord AC, equal to the radius of O are constructed. Find the degree measure of an angle between AB and AC.