7.3 Coordinate Geometry: Slope, Distance, Midpoint & Linear Graphs

7.3 Coordinate Geometry: Slope, Distance, Midpoint & Linear Graphs

Coordinate geometry bridges algebra and geometry by representing geometric shapes and linear relationships on a two-dimensional grid. On the AFQT Mathematics Knowledge subtest, coordinate geometry questions test your ability to work with linear equations, calculate slope, determine line intersections, and compute distance and midpoint values between coordinate pairs.

1. The Cartesian Coordinate Plane

The Cartesian coordinate plane is formed by two perpendicular number lines that intersect at the origin $(0,0)$:

• Horizontal axis: $x$-axis (positive to the right, negative to the left).
• Vertical axis: $y$-axis (positive upward, negative downward).

The plane is divided into four quadrants, numbered counterclockwise:

• Quadrant I: $(+x, +y)$ — Top-right
• Quadrant II: $(-x, +y)$ — Top-left
• Quadrant III: $(-x, -y)$ — Bottom-left
• Quadrant IV: $(+x, -y)$ — Bottom-right

Points are written as ordered pairs $(x, y)$, where $x$ represents the horizontal displacement from the origin and $y$ represents the vertical displacement.

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2. Slope of a Line ($m$)

The slope of a line measures its steepness and direction. It is defined as the ratio of vertical change (rise) to horizontal change (run) between any two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ on the line:

$m = \frac{\text{Rise}}{\text{Run}} = \frac{y_2 - y_1}{x_2 - x_1}$

The Four Types of Slope

Worked Example 1: Calculating Slope

Find the slope of the line passing through points $A(-3, 4)$ and $B(5, -2)$.

1. Label coordinates: $(x_1, y_1) = (-3, 4)$ and $(x_2, y_2) = (5, -2)$.
2. Apply the slope formula: $m = \frac{-2 - 4}{5 - (-3)} = \frac{-6}{5 + 3} = \frac{-6}{8} = -\frac{3}{4}$
3. The slope is $-\frac{3}{4}$ (a negative slope slanting downward).

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3. Parallel and Perpendicular Lines

Analyzing relationships between two lines is a frequent AFQT question type.

Parallel Lines

Two non-vertical lines are parallel if and only if they have the exact same slope: $m_1 = m_2$ Parallel lines never intersect.

Perpendicular Lines

Two non-vertical lines are perpendicular if and only if their slopes are opposite reciprocals (their product equals $-1$): $m_1 \cdot m_2 = -1 \implies m_2 = -\frac{1}{m_1}$ Perpendicular lines intersect at a right angle ($90^\circ$).

Worked Example 2: Parallel and Perpendicular Slopes

Given Line 1 with equation $y = \frac{2}{5}x + 7$:

• Any line parallel to Line 1 has slope $m = \mathbf{\frac{2}{5}}$.
• Any line perpendicular to Line 1 has slope $m = \mathbf{-\frac{5}{2}}$.

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4. Forms of Linear Equations

Linear equations can be expressed in three primary standard formats on the AFQT.

1. Slope-Intercept Form ($y = mx + b$)

This is the most useful form for graphing and identifying line properties: $y = mx + b$

• $m$: Slope of the line.
• $b$: $y$-intercept (the point where the line crosses the $y$-axis, $(0, b)$).

Finding Intercepts:

• $y$-intercept: Set $x = 0$ and solve for $y$ (equals $b$).
• $x$-intercept: Set $y = 0$ and solve for $x$ (equals $-\frac{b}{m}$).

2. Point-Slope Form ($y - y_1 = m(x - x_1)$)

Used to construct the equation of a line when given a point $(x_1, y_1)$ and the slope $m$: $y - y_1 = m(x - x_1)$

Worked Example 3: Writing a Line Equation

Find the equation of the line passing through point $(4, -1)$ with slope $m = 3$, in slope-intercept form.

1. Substitute into point-slope form: $y - (-1) = 3(x - 4) \implies y + 1 = 3x - 12$
2. Isolate $y$: $y = 3x - 13$
3. The slope-intercept form is $y = 3x - 13$.

3. Standard Form ($Ax + By = C$)

Where $A, B, C$ are integers (preferably $A \geq 0$).

• Slope $m = -\frac{A}{B}$.
• $y$-intercept $= \frac{C}{B}$.

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5. The Distance Formula

The distance formula calculates the straight-line distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ on the coordinate plane. It is derived directly from the Pythagorean Theorem ($a^2 + b^2 = c^2$):

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Worked Example 4: Calculating Distance

Find the distance between points $P(1, 2)$ and $Q(7, 10)$.

1. Assign coordinates: $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (7, 10)$.
2. Calculate horizontal change ($\Delta x$) and vertical change ($\Delta y$): $x_2 - x_1 = 7 - 1 = 6$ $y_2 - y_1 = 10 - 2 = 8$
3. Substitute into distance formula: $d = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$
4. The distance is $10$ units.

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6. The Midpoint Formula

The midpoint formula finds the point $M(x_m, y_m)$ located exactly halfway between two endpoints $(x_1, y_1)$ and $(x_2, y_2)$. It is simply the average of the $x$-coordinates and the average of the $y$-coordinates:

$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Worked Example 5: Finding the Midpoint

Find the midpoint $M$ of the segment connecting $A(-6, 8)$ and $B(4, 2)$.

1. Average the $x$-coordinates: $x_m = \frac{-6 + 4}{2} = \frac{-2}{2} = -1$
2. Average the $y$-coordinates: $y_m = \frac{8 + 2}{2} = \frac{10}{2} = 5$
3. The midpoint coordinates are $M(-1, 5)$.

Worked Example 6: Finding an Endpoint Given Midpoint

The midpoint of segment $CD$ is $M(3, 4)$. If endpoint $C$ is at $(1, 1)$, find the coordinates of endpoint $D(x_2, y_2)$.

1. Use midpoint $x$-equation: $\frac{1 + x_2}{2} = 3 \implies 1 + x_2 = 6 \implies x_2 = 5$
2. Use midpoint $y$-equation: $\frac{1 + y_2}{2} = 4 \implies 1 + y_2 = 8 \implies y_2 = 7$
3. Endpoint $D$ is at $(5, 7)$.

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7. Summary of Key Formulas for AFQT Coordinate Geometry

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8. AFQT Tips & Avoidable Mistakes

• Coordinate Order Consistency: When calculating slope $m = \frac{y_2 - y_1}{x_2 - x_1}$, make sure the same point's coordinates come first in both the numerator and denominator.
• Minus Sign Traps in Distance: Subtracting negative coordinates produces addition: $(x_2 - (-3))^2 = (x_2 + 3)^2$.
• Pythagorean Triples Shortcut: Notice that distance calculation $\sqrt{6^2 + 8^2} = 10$ is based on the 3-4-5 right triangle ratio (scaled by 2). Recognizing standard triples ($3\text{-}4\text{-}5$, $5\text{-}12\text{-}13$, $8\text{-}15\text{-}17$) saves precious calculation time on the test!