Notes
Angles and arcs
- Central angle = its arc. Inscribed angle = half its arc. An angle inscribed in a semicircle is $90°$.
- Angle inside a circle: half the sum of the two arcs. Angle outside: half the difference of the far and near arcs. Tangent-chord angle: half its arc.
Chords and tangents
- A radius perpendicular to a chord bisects it. Crossing chords: $ab = cd$.
- A tangent is perpendicular to the radius at the point of tangency. Tangents from one outside point are congruent.
Measures and equations
- Arc length $\displaystyle = \frac{\theta}{360} \cdot 2\pi r$. Sector area $\displaystyle = \frac{\theta}{360} \cdot \pi r^2$.
- Circle: $(x - h)^2 + (y - k)^2 = r^2$. Use completing the square to find the center and radius from a general equation.
Topics in this unit
Each topic has its own short notes and practice.
Practice
An inscribed angle is half its arc.
$\displaystyle \sqrt{17^2 - 8^2} = \sqrt{225} = 15$.
$\displaystyle \frac{120}{360}(2\pi \cdot 9) = \frac{1}{3}(18\pi) = 6\pi$.
$\displaystyle \frac{150}{360}(36\pi) = 15\pi$.
$3^2 + (-4)^2 = 9 + 16 = 25$.
Outside angle $\displaystyle = \frac{1}{2}(250 - 110) = 70°$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A circle has center $C(1, 2)$ and passes through the point $P(4, 6)$.
(a) Find the radius.
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$\displaystyle r = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{9 + 16} = 5$.
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(b) Write the equation of the circle.
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$(x - 1)^2 + (y - 2)^2 = 25$.
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(c) Find the length of a $72°$ arc and the area of the $72°$ sector. Give exact answers.
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Arc length $\displaystyle = \frac{72}{360}(2\pi \cdot 5) = \frac{1}{5}(10\pi) = 2\pi$. Sector area $\displaystyle = \frac{1}{5}(25\pi) = 5\pi$.
Check off the points you earned: