How can we find the measures of angles formed by intersecting lines in circles?
1. Angle formed by a tangent and a chord (or secant) is equal to half the measure of the
intercepted arc.
a. x=½a
2. Angles formed by intersecting chords are equal to half the sum of the arcs intercepted
by the angle and its vertical angle.
a. x= ½(a+b)
3. Angle formed by tangent and secant is equal to half the difference of the intercepted
arcs.
a. x=½(a-b)
How can we find the lengths of segments formed by intersecting lines in circles?
1. Intersecting Chords: product of the segments of 1 chord=products of the segments of
the other chord.
a. (AE)(BE)=(CE)(DE)
2. Two Secant Segments: Product of a whole secant segment and external
segment=Product of other whole secant and external segment.
a. (AC)(BC)=(EC)(DC)
3. Tangent Segment and Secant Segment:
a. On the circle, half the intercepted arc.
b. In the circle, half the sum
c. Outside the circle, half the difference
4. Tangent Segment Squared: Product of whole secant segment and external segment.
a. (AB)2=(BD)(BC)
How can we write equations of circles and graph circles?
➢ All circles are similar, there are 2 things that differ from circle to circle
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Radius length
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Location of center
➢ Center=(h,k)
➢ Point on the circle=(x,y)
r=√(x-h)2+(y-k)2
r2=(x-h)2+(y-k)2
How can we convert the equation of a circle from general to standard form?
➢ Standard form(center radius) for the equation of a circle
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w/ center at (h,k) and radius r
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(x-h)2+(y-k)2=r2
➢ The “general form” comes from expanding or multiplying the binomials in the equation.
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EX: (x-3)2+(y-2)2=81
x2-6x+9+y2-4y+4=81
x2+y2-6x-4y-68=0
➢ Completing the square is a process to turn a quadratic into a perfect square trinomial.
○
To find the number to complete the square, find (b/2)2 when the quadratic has
an a=1.
Circle Theorems:
1. In a circle, a radius perpendicular to a chord bisects the chord.
2. In a circle, a radius that bisects a chord is perpendicular to the chord.
3. In a circle, the perpendicular bisector of a chord passes through the center of the circle
4. In a circle, or congruent circles, congruent chords are equidistant from the center.
5. In a circle, or congruent circles, chords equidistant from the center are congruent.
6. In a circle, a pair of parallel chords intercept congruent arcs.
7. If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of
tangency.
8. Tangent segments to a circle from the same external point are congruent.
9. In a circle, or congruent circles, congruent central angles intercept congruent arcs.
10. In a circle, or congruent circles, congruent central angles intercept congruent chords.
11. In a circle, or congruent circles, congruent chords subtend congruent central angles.
12. In a circle, or congruent circles, congruent chords intercept congruent arcs.
13. An angle inscribed in a semi circle is a right angle
14. The opposite angles in an inscribed quadrilateral are supplementary.