Coordinate Geometry Practice Quiz with a Step-by-Step Interactive Lesson
Use the question set below to practice coordinate geometry (also called analytic geometry) on the coordinate plane / Cartesian plane: identifying ordered pairs, reading quadrants, graphing and interpreting points on the coordinate grid, calculating slope (gradient), writing the equation of a line (slope-intercept form, point-slope form, and standard form), finding x-intercepts and y-intercepts, working with parallel lines and perpendicular lines, and using the distance formula and midpoint formula for line segments. If you want a refresher, click Start lesson to open a step-by-step guide with worked examples and quick checks.
Answer the question set and review your mistakes at the end.
How this coordinate geometry practice works
1. Take the practice set: answer the coordinate geometry questions below.
2. Open the lesson (optional): review the coordinate plane, slope, line equations, intercepts, distance, midpoint, and parallel/perpendicular rules.
3. Retry: return to the question set and apply the coordinate geometry formulas immediately.
What you will learn in the coordinate geometry lesson
Coordinate plane foundations
Ordered pairs \((x,y)\), the origin, and the x-axis / y-axis
Quadrants and sign patterns for \((x,y)\)
Reflections across the x-axis and y-axis
Slope (gradient) & line direction
Slope formula \(m=\dfrac{y_2-y_1}{x_2-x_1}\) and rise over run
Special cases: horizontal lines (slope \(0\)) and vertical lines (slope undefined)
Reading steepness and direction from a slope value
Equations of lines & intercepts
Slope-intercept form \(y=mx+b\) and the y-intercept \((0,b)\)
Point-slope form \(y-y_1=m(x-x_1)\) from a point and slope
Finding x-intercepts and y-intercepts from standard form \(Ax+By=C\)
Distance, midpoint & line relationships
Distance formula and the Pythagorean theorem connection
Midpoint formula for segments on the coordinate plane
Parallel vs perpendicular lines (same slope vs negative reciprocals)
Purpose: Build a clear understanding of coordinate geometry (analytic geometry) so you can work confidently on the coordinate plane: plot points, read quadrants, calculate slope, write the equation of a line, find intercepts, and use the distance and midpoint formulas.
Success criteria
Identify the origin, x-axis, y-axis, and the quadrants on the coordinate plane.
Plot and interpret ordered pairs \((x,y)\) correctly.
Find the slope (gradient) of a line using two points.
Recognize the slope of horizontal and vertical lines.
Write an equation of a line in slope-intercept form \(y=mx+b\) and point-slope form \(y-y_1=m(x-x_1)\).
Find x-intercepts and y-intercepts from an equation.
Use slope rules for parallel and perpendicular lines.
Use the distance formula and midpoint formula for line segments on the coordinate plane.
Use reflections across axes: across x-axis \((x,y)\to(x,-y)\) and across y-axis \((x,y)\to(-x,y)\).
Key vocabulary
Coordinate plane (Cartesian plane): a grid formed by perpendicular number lines (x-axis and y-axis).
Ordered pair: \((x,y)\), where \(x\) is the horizontal coordinate and \(y\) is the vertical coordinate.
Origin: the point \((0,0)\) where the axes meet.
Quadrant: one of the four regions of the plane divided by the axes (I, II, III, IV).
Slope (gradient): \(m=\dfrac{\Delta y}{\Delta x}\), the rate of change of \(y\) with respect to \(x\).
y-intercept: where a line crosses the y-axis; in \(y=mx+b\), it is \(b\).
Parallel lines: lines with the same slope.
Perpendicular lines: lines meeting at \(90^\circ\); slopes are negative reciprocals (when defined).
Quick pre-check
Pre-check 1: In which quadrant is the point \((-2,3)\)?
Hint: Quadrant II has \(x<0\) and \(y>0\).
Pre-check 2: What is the slope of the line through \((1,1)\) and \((4,4)\)?
Hint: Use \(m=\dfrac{y_2-y_1}{x_2-x_1}\).
Coordinate Plane
The coordinate plane and plotting points
Learning goal: Read ordered pairs, identify quadrants, and apply reflection rules on the coordinate plane.
Key idea
The coordinate plane is formed by the x-axis (horizontal) and y-axis (vertical). A point is written as an ordered pair \((x,y)\). The sign of \(x\) and \(y\) tells you the quadrant. Reflections are quick coordinate changes:
Across the x-axis: \((x,y)\to(x,-y)\)
Across the y-axis: \((x,y)\to(-x,y)\)
Worked example
Example: Plot \((-2,3)\) and reflect it across the x-axis.
\((-2,3)\) means 2 left and 3 up, so it lies in Quadrant II. Across the x-axis, only the \(y\)-value changes sign: \[ (-2,3)\to(-2,-3). \]
Try it
Try it 1: What is the reflection of \((-2,4)\) across the x-axis?
Hint: Across the x-axis, \(x\) stays the same and \(y\) changes sign.
Try it 2: What is the equation of the vertical line through \((-3,2)\)?
Hint: A vertical line keeps \(x\) constant for every point on the line.
Summary
Coordinates are ordered pairs \((x,y)\): move \(x\) horizontally, then \(y\) vertically.
Reflections across axes flip the sign of one coordinate.
Slope
Slope (gradient) and rate of change
Learning goal: Calculate slope from two points and recognize special slopes for vertical and horizontal lines.
Key idea
The slope \(m\) of a line through \((x_1,y_1)\) and \((x_2,y_2)\) is: \[ m=\frac{y_2-y_1}{x_2-x_1}. \] Slope measures how much \(y\) changes for each 1 unit change in \(x\) (rise over run). A horizontal line has slope \(0\). A vertical line has undefined slope because \(x_2-x_1=0\).
Worked example
Example: What is the slope of the line through \((-2,5)\) and \((4,-1)\)?
\[ m=\frac{-1-5}{4-(-2)}=\frac{-6}{6}=-1. \] A slope of \(-1\) means that when \(x\) increases by 1, \(y\) decreases by 1.
Try it
Try it 1: What is the slope of the line through \((1,2)\) and \((3,6)\)?
Hint: \(m=\dfrac{6-2}{3-1}\).
Try it 2: What is the slope of the vertical line \(x=5\)?
Hint: For a vertical line, \(\Delta x=0\), so \(\dfrac{\Delta y}{\Delta x}\) is not defined.
Summary
Slope formula: \(m=\dfrac{y_2-y_1}{x_2-x_1}\).
Horizontal slope \(=0\); vertical slope is undefined.
Line Equations
Equation of a line: slope-intercept and point-slope
Learning goal: Write line equations using slope-intercept form and point-slope form, and recognize parallel lines.
Key idea
The most common line forms are:
Slope-intercept form: \(\;y=mx+b\) where \(m\) is slope and \(b\) is the y-intercept.
Point-slope form: \(\;y-y_1=m(x-x_1)\) using a point \((x_1,y_1)\) and slope \(m\).
Parallel lines have the same slope. If you know the slope and one point, point-slope form is a reliable starting point.
Worked example
Example: Find the equation of the line through \((-2,1)\) and \((2,3)\).
First find the slope: \[ m=\frac{3-1}{2-(-2)}=\frac{2}{4}=\frac{1}{2}. \] Use point-slope form with \((-2,1)\): \[ y-1=\frac{1}{2}(x+2). \] Simplify to slope-intercept form: \[ y=\frac{1}{2}x+2. \]
Try it
Try it 1: What is the equation of the line with slope \(3\) through the origin?
Hint: Through the origin means \(b=0\) in \(y=mx+b\).
Try it 2: What is the equation of the line parallel to \(y=5x-2\) through \((0,3)\)?
Hint: Parallel lines have the same slope. Use \((0,3)\) to find \(b\).
Summary
Use \(y=mx+b\) when you know slope and y-intercept (or can find them).
Use \(y-y_1=m(x-x_1)\) when you know slope and a point.
Parallel lines share the same slope.
Intercepts
Intercepts and standard form \(Ax+By=C\)
Learning goal: Find x-intercepts and y-intercepts quickly and interpret them correctly on the coordinate plane.
Key idea
To find intercepts, use these reliable "set to zero" rules:
y-intercept: set \(x=0\).
x-intercept: set \(y=0\).
In slope-intercept form \(y=mx+b\), the y-intercept is \(b\). In standard form \(Ax+By=C\), it is often fastest to substitute \(x=0\) or \(y=0\).
Worked example
Example: What is the y-intercept of the line \(y=-\tfrac{3}{4}x+2\)?
In \(y=mx+b\), the y-intercept is \(b\). Here \(b=2\), so the y-intercept is \(2\) (the point \((0,2)\)). (Extra check: set \(x=0\), so \(y=2\).)
Try it
Try it 1: What is the x-intercept of the line \(3x+4y=12\)?
Hint: For the x-intercept, set \(y=0\) and solve for \(x\).
Try it 2: What is the y-intercept of the line \(4x+3y=12\)?
Hint: For the y-intercept, set \(x=0\) and solve for \(y\).
Summary
y-intercept: set \(x=0\). x-intercept: set \(y=0\).
In \(y=mx+b\), the y-intercept is \(b\).
Parallel & Perpendicular
Parallel and perpendicular lines
Learning goal: Use slope rules to identify and create parallel and perpendicular lines in coordinate geometry.
Key idea
Two non-vertical lines are:
Parallel if they have the same slope.
Perpendicular if their slopes are negative reciprocals, so \(m_1m_2=-1\).
Special cases: a vertical line is perpendicular to a horizontal line. (Vertical slope is undefined, horizontal slope is 0.)
Worked example
Example: What is the slope of a line perpendicular to \(y=\tfrac{3}{4}x+1\)?
The given slope is \(\tfrac{3}{4}\). The negative reciprocal is: \[ -\frac{4}{3}. \] So any line perpendicular to \(y=\tfrac{3}{4}x+1\) has slope \(-\tfrac{4}{3}\).
Try it
Try it 1: What is the slope of a line perpendicular to \(y=2x+1\)?
Hint: Take the negative reciprocal of \(2\).
Try it 2: What is the equation of the horizontal line through \((0,-2)\)?
Learning goal: Use distance and midpoint formulas accurately on the coordinate plane.
Key idea
The distance between \((x_1,y_1)\) and \((x_2,y_2)\) is: \[ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. \] The midpoint of the segment joining the two points is: \[ M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right). \]
Worked example
Example: What is the distance between \((0,0)\) and \((3,4)\)?
Line equations: \(y=mx+b\) and \(y-y_1=m(x-x_1)\).
Intercepts: set \(x=0\) for y-intercept; set \(y=0\) for x-intercept.
Parallel: same slope. Perpendicular: negative reciprocal slopes (when defined).
Distance and midpoint: \(\sqrt{(\Delta x)^2+(\Delta y)^2}\) and \(\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\).
Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the coordinate geometry skill you need.
Practice set
Coordinate Geometry practice questions with instant score
Answer all 10 questions below, then get your final score and a mistake review at the end so you know exactly what to improve.
0/10answered
Question 1Not answered
What are the coordinates of the origin in the Cartesian plane?
Correct answer: C. \((0,0)\)
Explanation: The origin is where the \(x\)- and \(y\)-axes intersect: \((0,0)\).
Question 2Not answered
What is the equation of the line through \((2,3)\) and \((5,7)\) in slope–intercept form?
Correct answer: C. \(y = \frac{4}{3}x + \frac{1}{3}\)
