ACT Math 5: Coordinate Geometry — Lines and Slopes

How can the Euclidean distance between $(x_1,y_1)$ and $(x_2,y_2)$ be calculated?

Use $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$, which follows from the Pythagorean theorem.

What is the midpoint of two points $(x_1,y_1)$ and $(x_2,y_2)$?

The midpoint is $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. It averages the corresponding coordinates.

What is the slope of a nonvertical line through $(x_1,y_1)$ and $(x_2,y_2)$?

The slope is $m=\frac{y_2-y_1}{x_2-x_1}$, or rise divided by run, provided $x_2\ne x_1$.

Cartesian coordinate system

A system that uniquely identifies each point using signed numerical coordinates measured relative to fixed, mutually perpendicular oriented axes. In two dimensions, a point is represented by an ordered pair $(x,y)$.

What information defines a standard two-dimensional Cartesian coordinate system?

It consists of two perpendicular axes intersecting at the origin, a common unit of length, and an orientation for each axis.

Origin

The point where the coordinate axes intersect. Its coordinates are $(0,0)$.

How are the coordinates of a point $(x,y)$ determined in a Cartesian plane?

Project the point perpendicularly onto each axis. The signed position on the first axis is $x$, and the signed position on the second axis is $y$.

How do the signs of Cartesian coordinates identify a quadrant?

Quadrant I has $(+,+)$, quadrant II has $(-,+)$, quadrant III has $(-,-)$, and quadrant IV has $(+,-)$. The conventional numbering proceeds counterclockwise from quadrant I.

How can the distances from $(x,y)$ to the coordinate axes be found?

The distance to the $x$-axis is $|y|$, and the distance to the $y$-axis is $|x|$.

What is a Cartesian plane?

A Euclidean plane equipped with a chosen two-dimensional Cartesian coordinate system.

What does the sign of a line's slope indicate?

A positive slope means the line rises from left to right, a negative slope means it falls, and a zero slope means it is horizontal.

What is the slope of a vertical line?

A vertical line has undefined slope because its run is zero, so the slope formula would require division by zero.

What is the slope-intercept form of a line?

The slope-intercept form is $y=mx+b$, where $m$ is the slope and $b$ is the $y$-intercept.

What is the point-slope form of a line?

A line with slope $m$ through $(x_1,y_1)$ can be written as $y-y_1=m(x-x_1)$.

What is the standard form of a linear equation?

Standard form is $Ax+By=C$, where $A$, $B$, and $C$ are constants, typically written with integer coefficients and $A\ge 0$.

How can the slope and intercepts be identified from $Ax+By=C$?

If $B\ne0$, rewrite it as $y=-\frac{A}{B}x+\frac{C}{B}$. Thus the slope is $-\frac{A}{B}$ and the $y$-intercept is $\frac{C}{B}$. The $x$-intercept is found by setting $y=0$.

How do you find the equation of a line given its slope and one point?

Substitute the slope $m$ and point $(x_1,y_1)$ into point-slope form: $y-y_1=m(x-x_1)$. Simplify into the requested form if needed.

How do you find the equation of a line through two points?

First calculate $m=\frac{y_2-y_1}{x_2-x_1}$ when $x_1\ne x_2$. Then use either point in $y-y_1=m(x-x_1)$.

What is the equation of a horizontal line?

A horizontal line has slope $0$ and an equation of the form $y=c$, where $c$ is its constant $y$-coordinate.

What is the equation of a vertical line?

A vertical line has undefined slope and an equation of the form $x=c$, where $c$ is its constant $x$-coordinate.

How are the slopes of parallel lines related?

Distinct nonvertical parallel lines have equal slopes. Vertical lines are parallel to one another and all have undefined slope.

How are the slopes of perpendicular nonvertical lines related?

Their slopes are negative reciprocals: $m_1m_2=-1$. For example, a line with slope $\frac{2}{3}$ is perpendicular to one with slope $-\frac{3}{2}$.

How can you determine whether two lines are parallel or perpendicular from their equations?

Compare their slopes after rewriting the equations in slope-intercept form. Equal slopes indicate parallel lines; slopes whose product is $-1$ indicate perpendicular lines, with vertical and horizontal lines treated as perpendicular.

How can a line be translated in the Cartesian plane?

Translating every point by $(a,b)$ changes $(x,y)$ to $(x+a,y+b)$. This preserves distances and directions, so a line remains a parallel line.