Geometry: Circles

Circles, the perfect shape! On this page we hope to clear up problems that you might have with circles and the figures, such as radii, associated with them. Just start reading or follow any of the links below to start understanding circles better!

Chords & radii
Tangents
Congruent arcs
Inscribed angles
Angles involving tangents and/or secants
Segments in circles
Circumference and arc legth
Quiz on circles

Chords & Radii

All the "parts" of a circle, such as the radius, the diameter, etc., have a relationship with the circle or another "part" that can always be expressed as a theorem. The two theorems that deal with chords and radii (plural of radius) are outlined below.

1. If a radius of a circle is perpendicular to a chord, then the radius bisects the chord. Example: Example Figure

2. In a circle or in congruent circles, if two chords are the same distance from the center, then they are congruent.

Using these theorems in action is seen in the example below.

1.  Problem: Find CD in the accompanying figure.
      Given: Circle R is congruent to circle S.
             Chord AB = 8.
             RM = SN.
   Solution: By theorem number 2 above, segment AB is congruent
             to segment CD.  Therefore, CD equals 8.

Tangents

Oh, the wonderfully confusing world of geometry! :-) The tangent being discussed here is not the trigonometric ratio. This kind of tangent is a line or line segment that touches the perimeter of a circle at one point only and is perpendicular to the radius that contains the point.

Example:

1.  Problem: Find the value of x in the accompanying figure.
      Given: Segment AB is tangent to circle C at B.
   Solution: x is a radius of the circle.  Since x
             contains B, and AB is a tangent segment, x
             must be perpendicular to AB (the definition of a tangent
             tells us that).

             If it is perpendicular, the triangle formed by x, AB,
             and CA is a right triangle.

             Use the Pythagorean Theorem to
             solve for x.

             15^2 + x^2 = 17^2
             x^2 = 64
             x = 8

Congruent Arcs

Congruent arcs are arcs that have the same degree measure and are in the same circle or in congruent circles.

Arcs are very important and let us find out a lot about circles. Two theorems involving arcs and their central angles are outlined below.

1. For a circle or for congruent circles, if two minor arcs are congruent, then their central angles are congruent.

2. For a circle or for congruent circles, if two central angles are congruent, then their arcs are congruent. Example: Example Figure

Inscribed Angles

An inscribed angle is an angle with its vertex on a circle and with sides that contain chords of the circle. This figure shows an inscribed angle.

The most important theorem dealing with inscribed angles is stated below.

The measure of an inscribed angle is equal to one-half the degree measure of its intercepted arc.

Example:

1.  Problem: Find the measure of each arc or angle listed below (and
             shown in the accompanying figure.
             
             arc QSR     angle Q     angle R
   Solution: Arc QSR is 180° because it is twice the
             measure of its inscribed angle (angle QPR, which is
             90°).

             Angle Q is 60° because it is half of its
             intercepted arc, which is 120°.

             Angle R is 30° by the Triangle Sum Theorem
             which says a triangle has three angles which have measures
             that equal 180° when added together.

In the last problem's figure, you noticed that angle P is inscribed in semicircle QPR and angle P = 90°. This leads us to our next theorem, which is stated below.

Any angle inscribed in a semicircle is a right angle.

The one last theorem dealing with inscribed angles is a bit more complicated because it deals with quadrilaterals, too. It is stated below.

If a quadrilateral is inscribed in a circle, then both pairs of opposite angles are supplementary.

Example:

1.  Problem: Find the measure of arc GDE in the
             accompanying figure.
   Solution: By the theorem stated above, angle D and angle F
             are supplementary.  Therefore, angle F equals 95°.
             The first theorem discussed in this section tells us the
             measure of an arc is twice that of its inscribed angle.
             With that theorem, arc GDE is 190°.

Angles Involving Tangents and/or Secants

When two secants intersect inside a circle, the measure of each angle formed is related to one-half the sum of the measures of the intercepted arcs. This figure shows this theorem in action.

In the figure, arc AB and arc CD are 60° and 50°, respectively. By the above stated theorem, the measures of both angle 1 and angle 2 in the figure are 55°.

Sometimes, secants intersect outside of circles. When this happens, the measure of the angle formed is equal to one-half the difference of the degree measures of the intercepted arcs.

Example:

1.  Problem: Find the measure of angle 1 in the
             accompanying figure.
     Givens: Arc AB = 60°
             Arc CD = 100°
   Solution: By the theorem stated above, the measure of
             angle 1 = .5((arc CD) - (arc AB)).

             angle 1 = .5((100 - 60))
             angle 1 = 20°

Line Segments Created by Intersecting Secants and/or Tangents

Another way secants can intersect in circles is if they are only in line segments. There is a theorem that tells us when two chords intersect inside a circle, the product of the measures of the two segments of one chord is equal to the product of the measures of the two segments of the other chord. In the accompanying figure, chords PR and QS intersect. By the theorem stated above, PT * TR = ST * TQ.

Circumference and Arc Length

One last thing that has to be discussed when dealing with circles is circumference, or the distance around a circle. The circumference of a circle equals 2 times PI times the measure of the radius. The postulate is usually represented by the following equation (where C represents the circumference and r stands for radius): C = 2(PI)r.

For example, if a circle has a radius of 3, the circumference of the circle is 6(PI).

Also, you can find the length of any arc when you know its degree measure and the measure of a radius with the following formula (L = length, n = degree measure of arc, r = radius): L = (n/360)(2(PI)r).

Example:

1.  Problem: Find the length of a 24° arc of a circle with a 5 cm
             radius.  Accompanying figure.

   Solution:      n             24           2(PI)
             L = ---(2(PI)r) = ---(2(PI))5 = -----
                 360           360             3

             The length of the arc is (2/3)(PI) cm.

Take the quiz on circles. The quiz is very useful for either review or to see if you've really got the topic down.

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