Chapter 4

Properties of CirclesPart 1

Definition: the set of all points equidistant from a central point.

Terms You Should Know!

Radius: a line segment joining the centre of the circle to a

point on the circle

All radii of a circle are equal in length.

Diameter: a chord passing through the

centre of a circle

The diameter of a circle is twice the length of the radius.

Chord: a line segment joining two points on the circle

The diameter is the longest chord in a circle.

Semi-Circle: half of a circle

Circumference of a Circle:

the distance around a circle; its perimeter.

Arc: part of the circumference of a circle

Minor Arc: less than half the circumference

Major Arc: more than half the circumference

Interior Points: A point is inside a circle if its distance from the center is less than the radius, inside the circumference.

INTERIOR

EXTERIOR

Exterior Point: A point is outside a circle if its distance from the center is more than the radius, outside the circumference.

The part of the secant on the interior of the circle is called a chord.

Secant: a line cutting through the circumference of a circle at two points.

Tangent: a line touching the circumference

of the circle at only one point.

No part of the tangent is in the interior of the circle.

The point where the tangent touches the circle is called the point of tangency.

Perpendicular Bisector: a line that bisects a line segment and forms a

900 angle.

Concentric Circles: circles that share the

same center.

Isosceles Triangle:

a triangle with two equal sides and two equal angles.

Equilateral Triangle:

a triangle with all sides equal.

Equilateral triangles are also equiangular. All angles measure 600.

Transversal Parallel Lines Theorem· If a transversal intersects two parallel lines, the alternate angles equal.· If a transversal intersects two parallel lines, the corresponding angles equal.

Vertical Angles: two nonadjacent angles formed by two intersecting lines.

Vertical Angles are Congruent

Ellipse: a set of points whose sum of its distances from two fixed points is constant, forming an oval shape.

Converse: a statement formed by interchanging the “if” and “then” parts of the original statement.

Ex: Statement: If a figure is a triangle, then it is a polygon.

Converse: If a figure is a polygon, then it is a triangle.

The converse of a statement is not necessarily true.

Example:Statement: If a triangle is equiangular, then it is equilateral.

Converse: If a triangle is equilateral, then it is equiangular.

Since both the statement and its converse are true, the two statements can be written as:

A triangle is equiangular iff it is equilateral.

“iff” means“if and only if”

What is the converse of:“If a triangle is inscribed in a semi-circle, then the triangle is a right triangle.”?

Question

(A) If a triangle is inscribed in a semi-circle, then the triangle is not a right triangle.(B) If a triangle is not inscribed in a semi-circle, then the triangle is not a right triangle.(C) If a triangle is a right triangle, then it is inscribed in a semi-circle.(D) If a triangle is a right triangle, then it cannot be inscribed in a semi-circle.

QuestionWhat is the converse of:“If two chords of a circle are parallel, then the two arcs between the chords are congruent” ?

(A) If the two arcs between the chords in a circle are congruent, then the chords arenot parallel.(B) If the two arcs between the chords in a circle are not congruent, then the chordsare not parallel.(C) If the two arcs between the chords in a circle are congruent, then the chords areparallel.(D) If two chords of a circle are not parallel, then the arcs between the chords are notcongruent.

Workbook

Page 217Questions 11-16