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 scrolling
down or click one of the links below to start understanding
Angles involving tangents and/or secants
Segments in circles
Circumference and arc length
Quiz on Circles
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. Problem: Find CD. 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.
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.
1. Problem: Find the value of x. 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. 152 + x2 = 172 x2 = 64 x = 8
Congruent arcs are arcs that have the same degree measure
and are in the same circle or in congruent circles.
An inscribed angle is an angle with its vertex on a circle and with sides that contain chords of the circle. The figure below shows an inscribed angle.
The most important theorem dealing with inscribed
angles is stated below.
1. Problem: Find the measure of each arc or angle listed below. arc QSR angle Q angle R Solution: Arc QSR is 180o because it is twice the measure of its inscribed angle (angle QPR, which is 90o). Angle Q is 60o because it is half of its intercepted arc, which is 120o. Angle R is 30o by the Triangle Sum Theorem which says a triangle has three angles which have measures that equal 180o when added together.
In the last problem's figure, you noticed that
angle P is inscribed in semicircle QPR
and angle P = 90o. This leads us
to our next theorem, which is stated below.
1. Problem: Find the measure of arc GDE. Solution: By the theorem stated above, angle D and angle F are supplementary. Therefore, angle F equals 95o. 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 190o.
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. The figure below shows this theorem in action.
1. Problem: Find the measure of angle 1. Givens: Arc AB = 60o Arc CD = 100o Solution: By the theorem stated above, the measure of angle 1 = .5((arc CD) - (arc AB)) angle 1 = .5((100 - 60)) angle 1 = 20o
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 figure below, chords PR and QS intersect. By the theorem stated above, PT * TR = ST * TQ.
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. That postulate
is usually represented by the following equation (where C represents circumference and
r stands for radius): C = 2(PI)r.
1. Problem: Find the length of a 24o arc of a circle with a 5 cm radius. 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. (Very useful to review or to see if you've really got this topic down.) Do it!