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.Back to top.
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 = 8Back to top.
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
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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.
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.
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°.Back to top.
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.
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°Back to top.
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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).
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.Back to top.
Take the quiz on circles. The quiz is very useful for either review or to see if you've really got the topic down.