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Coordinate Geometry

Distance, midpoint, slope, lines, circles, conics and transformations.

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Coordinate Geometry

Distance, midpoint, slope, lines, circles, conics and transformations.

📍 Coordinate Plane Basics

Part 1 of 7 — Plotting, Quadrants, Distance & Midpoint

The coordinate plane is a two-dimensional surface formed by the intersection of a horizontal number line (the x-axis) and a vertical number line (the y-axis). Every point is described by an ordered pair (x,y)(x, y).

QuadrantSignsExample
I(+,+)(+, +)(3,5)(3, 5)
II(−,+)(-, +)(−4,2)(-4, 2)
III(−,−)(-, -)(−1,−6)(-1, -6)
IV(+,−)(+, -)(7,−3)(7, -3)

Key facts:

  • Points on the x-axis have y=0y = 0.
  • Points on the y-axis have x=0x = 0.
  • The origin is (0,0)(0, 0).

📈 Slope & Linear Equations

Part 2 of 7 — Slope Formula, Slope-Intercept, Point-Slope, Parallel & Perpendicular

The slope of a line through (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:

m=y2−y1x2−x1=riserunm = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}

Slope TypeValueVisual
Positivem>0m > 0Rising left → right
Negativem<0m < 0Falling left → right
Zerom=0m = 0Horizontal line
Undefineda0\frac{a}{0}Vertical line

Linear equation forms:

  • Slope-intercept: y=mx+by = mx + b (slope mm, y-intercept bb)
  • Point-slope: y−y1=m(x−x1)y - y_1 = m(x - x_1)
  • Standard form: Ax+By=CAx + By = C

📊 Graphing Lines & Inequalities

Part 3 of 7 — Intercepts, Graphing Methods, Shading Regions

There are three standard ways to graph a line:

MethodWhat You Need
Slope-interceptSlope mm and y-intercept bb
Intercept methodx-intercept and y-intercept
Table of valuesPick xx-values, compute yy

Finding intercepts:

  • x-intercept: Set y=0y = 0 and solve for xx.
  • y-intercept: Set x=0x = 0 and solve for yy.

Example 1: 3x+2y=123x + 2y = 12

  • x-intercept: 3x=12  ⟹  x=43x = 12 \implies x = 4 → point (4,0)(4, 0)
  • y-intercept: 2y=12  ⟹  y=62y = 12 \implies y = 6 → point (0,6)(0, 6)

⭕ Circles on the Coordinate Plane

Part 4 of 7 — Standard Form, Center & Radius, Completing the Square

The standard form equation of a circle is:

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

  • Center: (h,k)(h, k)
  • Radius: rr
  • Diameter: 2r2r (so the radius is half the diameter)

If a problem gives the endpoints of a diameter, the center is their midpoint and the radius is half the distance between them.

EquationCenterRadius
(x−3)2+(y+1)2=16(x - 3)^2 + (y + 1)^2 = 16(3,−1)(3, -1)44
x2+y2=25x^2 + y^2 = 25(0,0)(0, 0)55
(x+2)2+(y−5)2=9(x + 2)^2 + (y - 5)^2 = 9(−2,5)(-2, 5)33

Key insight: Watch the signs! (y+1)(y + 1) means k=−1k = -1, and (x+2)(x + 2) means h=−2h = -2.

🔵 Conic Sections Overview

Part 5 of 7 — Parabola Vertex Form, Ellipses & Hyperbolas for the ACT

A conic section is a curve obtained by slicing a cone with a plane. The four types are:

ConicStandard FormShape
Circle(x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2Round
Parabolay=a(x−h)2+ky = a(x-h)^2 + kU-shaped
Ellipse(x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1Oval
Hyperbola(x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1Two branches

On the ACT, parabolas appear most often. Ellipses and hyperbolas are rare but worth recognizing.

🔄 Transformations on the Coordinate Plane

Part 6 of 7 — Translations, Reflections, Rotations & Dilations

A transformation changes a figure's position, size, or orientation. The four main types:

TransformationWhat ChangesPreserves Shape & Size?
TranslationPositionYes (rigid)
ReflectionOrientationYes (rigid)
RotationOrientation & positionYes (rigid)
DilationSizeNo (similar, not congruent)

Rigid motions (translation, reflection, rotation) preserve distances and angles.

🏆 Review & Mixed Practice

Part 7 of 7 — Formula Cheat Sheet & Mixed ACT Coordinate Geometry Problems

Here is your complete cheat sheet of coordinate geometry formulas for the ACT:

FormulaExpression
Distanced=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}
MidpointM=(x1+x22,y1+y22)M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)
Slopem=y2−y1x2−x1m = \frac{y_2-y_1}{x_2-x_1}
Slope-intercepty=mx+by = mx + b
Point-slopey−y1=m(x−x1)y - y_1 = m(x - x_1)
Circle(x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2
Parabola vertexy=a(x−h)2+ky = a(x-h)^2 + k
Parallel slopesm1=m2m_1 = m_2
Perpendicular slopesm1⋅m2=−1m_1 \cdot m_2 = -1

Strategy for ACT Coordinate Geometry:

  1. Identify what formula you need.
  2. Label known values clearly.
  3. Plug in and simplify.
  4. Watch for sign errors — they are one of the most common mistakes.
Explain using:
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📌 Related Topics in ACT Math

Pre-Algebra and Elementary AlgebraACT Algebra and FunctionsACT Geometry and TrigonometryIntermediate Algebra and Coordinate GeometryACT Statistics and ProbabilityPlane Geometry and Trigonometry

❓ Frequently Asked Questions

What is Coordinate Geometry?▾
Distance, midpoint, slope, lines, circles, conics and transformations.
How can I study Coordinate Geometry effectively?▾
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Regular review and active practice are key to retention.
Is this Coordinate Geometry study guide free?▾
Yes — all study notes, flashcards, and practice problems for Coordinate Geometry on Study Mondo are free to access. No account is needed.
What course covers Coordinate Geometry?▾
Coordinate Geometry is part of the ACT Prep course on Study Mondo, specifically in the ACT Math section. You can explore the full course for more related topics and practice resources.