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Circle Equations

Find center and radius instantly. Don't let factoring slow you down.

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Mastery
Geometry

Standard Circle Form

\[ (x-h)^2 + (y-k)^2 = r^2 \]

Center is \((h, k)\). Radius is \(r\).
Trap: Remember to switch the signs!

📝 Example: Basic Center

Find center and radius of \((x+2)^2 + (y-5)^2 = 49\).

Result: Center (-2, 5), Radius 7.

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Geometry

General Form Hack

\[ \text{Given: } x^2 + y^2 + Dx + Ey + F = 0 \]
\[ \text{Center: } (h,k) = (-D/2, -E/2) \]
\[ \text{Radius: } r = \sqrt{h^2 + k^2 - F} = \sqrt{\frac{D^2}{4} + \frac{E^2}{4} - F} \]

Halve the coefficients of \(x\) and \(y\), then flip their signs to find the center \((h,k)\). Then, plug \(h, k\), and \(F\) into the radius formula.

Practice: "Find center and radius of \(x^2 + y^2 - 10x + 6y - 2 = 0\)"

🚫 School Way (Rigorous)

✅ Practix Way (Optimal)

🎓 Theoretical Proof: Why These Formulas Work

Starting from General Form:

\( x^2 + y^2 + Dx + Ey + F = 0 \)

Complete the square for both \(x\) and \(y\):

\( (x^2 + Dx + \tfrac{D^2}{4}) + (y^2 + Ey + \tfrac{E^2}{4}) = -F + \tfrac{D^2}{4} + \tfrac{E^2}{4} \)

Which gives us the Standard Form:

\( (x + \tfrac{D}{2})^2 + (y + \tfrac{E}{2})^2 = \tfrac{D^2}{4} + \tfrac{E^2}{4} - F \)

Comparing with \((x - h)^2 + (y - k)^2 = r^2\), we read off:

\( h = -\tfrac{D}{2},\; k = -\tfrac{E}{2},\; r = \sqrt{h^2 + k^2 - F} \)
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Geometry

Arc Length (Degrees)

\[ L = 2\pi r (\frac{\theta}{360}) \]

It's just the circumference times the fraction of the circle.

📝 Example: Basic Arc

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.

Geometry
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Geometry

Sector Area (Degrees)

\[ A = \pi r^2 (\frac{\theta}{360}) \]

It's just the total area times the fraction of the circle.

Shortcut
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Geometry

Arc Length (Radians)

\[ s = r\theta \]

The definition of a radian! \(\theta\) must be in radians.

📝 Example: Radian Shortcut

Radius 5, Angle \(2\pi/5\). Find arc length.

\(s = 5 \cdot \frac{2\pi}{5} = 2\pi\)

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Geometry

Sector Area (Radians)

\[ A = \frac{1}{2}r^2 \theta \]

Much faster than converting to degrees.

Concept
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Geometry

Inscribed Angle

\[ \text{Angle} = \frac{1}{2} \text{Arc} \]

An angle on the edge is half the central angle (or half the arc).

📝 Example: Edge Angle

If an arc measure is \(100^\circ\), the inscribed angle is \(50^\circ\).

Concept
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Geometry

Central Angle

\[ \text{Angle} = \text{Arc} \]

An angle at the center equals the arc measure.

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Geometry

Tangent-Radius

\[ m\angle = 90^\circ \]

A radius connected to a tangent line always forms a right angle.

📝 Example: Right Triangle setup

Used to solve for missing lengths using the Pythagorean Theorem (\(a^2 + b^2 = c^2\)).

Process
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Geometry

Diameter to Equation

\[ \text{Midpoint} = \text{Center} \]

Given endpoints: 1. Find Midpoint (Center). 2. Find distance to center (Radius).

📝 Example: Endpoints to Circle

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: \((x-3)^2 + (y-4)^2 = 25\).

Rare
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Geometry

Cyclic Quadrilateral

\[ \text{Opposite Angles Sum} = 180^\circ \]

If a 4-sided shape is inside a circle, opposite angles add to 180.

📝 Example: Sum check

If \(\angle A = 70^\circ\), the opposite angle \(\angle C = 110^\circ\).

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Geometry

Area of Ring (Annulus)

\[ A = \pi R^2 - \pi r^2 \]

Big circle minus small circle.