Find center and radius instantly. Don't let factoring slow you down.
Center is \((h, k)\). Radius is \(r\).
Trap:
Remember to switch the signs!
Find center and radius of \((x+2)^2 + (y-5)^2 = 49\).
Result: Center (-2, 5), Radius 7.
When given \(x^2 + y^2 + Dx + Ey + F = 0\), just halve the coefficients and flip the sign to find the center.
\(x^2 + y^2 - 10x + 6y - 2 = 0\)
Center x: \(-10 \div 2 = -5 \rightarrow +5\)
Center y: \(6 \div 2 = 3 \rightarrow -3\)
Center is \((5, -3)\).
It's just the circumference times the fraction of the circle.
Radius 6, Angle \(60^\circ\). Find arc length.
\(L = 2\pi(6) \cdot \frac{60}{360} = 12\pi \cdot \frac{1}{6} = 2\pi\)
Result: 2π.
It's just the total area times the fraction of the circle.
The definition of a radian! \(\theta\) must be in radians.
Radius 5, Angle \(2\pi/5\). Find arc length.
\(s = 5 \cdot \frac{2\pi}{5} = 2\pi\)
Much faster than converting to degrees.
An angle on the edge is half the central angle (or half the arc).
If an arc measure is \(100^\circ\), the inscribed angle is \(50^\circ\).
An angle at the center equals the arc measure.
A radius connected to a tangent line always forms a right angle.
Used to solve for missing lengths using the Pythagorean Theorem (\(a^2 + b^2 = c^2\)).
Given endpoints: 1. Find Midpoint (Center). 2. Find distance to center (Radius).
Endpoints of a diameter: \((0, 0)\) and \((6, 8)\).
1. Midpoint = \((3, 4)\). This is the center.
2. Radius = distance from \((0, 0)\) to \((3, 4) = \sqrt{3^2 + 4^2} = 5\).
Result: \((x-3)^2 + (y-4)^2 = 25\).
If a 4-sided shape is inside a circle, opposite angles add to 180.
If \(\angle A = 70^\circ\), the opposite angle \(\angle C = 110^\circ\).
Big circle minus small circle.