Idiot\'s Guides Basic Math and Pre-Algebra

Chapter 15: Circles 201

draw two chords in the same circle and the two chords are the same length, then the arcs they
cut off will be the same size. Congruent chords cut off congruent arcs.

DEFINITION

A chord is a line segment that connects two points on a circle. The diameter is the

longest chord that can be drawn in a circle and passes through the center.

If you draw two radii, they both come from the center point, and they form a central angle. If
you draw two chords that come from the same point on the circle, you form an angle that also
intercepts an arc, but you can see that it’s smaller than the central angle that intercepts that same
arc. An inscribed angle is an angle whose sides are chords and whose vertex lies on the circle. In the
figure, central angle ’AOB and inscribed angle ’ACB both intercept arc AB, but you can see that
’ACB is smaller.

DEFINITION

An inscribed angle is an angle with its vertex on the circle whose sides are chords.

If you look at 'AOC, you can see it’s an isosceles triangle because radii AO = OC, and that means
that m’OAC = m’OCA. In addition, ’AOB is an exterior angle of the triangle, so m’AOB =
m’OAC + M’OCA = 2m’OCA. That means the central angle is twice the size of the inscribed
angle, or, in other words, the inscribed angle is half the size of the central angle. The measure of
an inscribed angle is equal to one-half the measure of its intercepted arc.

A

B

C

O