Geometry

• Find and use slope: read it from y = mx + b, compute it from two points, and use equal slopes for parallel lines and negative reciprocal slopes for perpendicular lines
• Calculate the distance between two points and the midpoint of a segment
• Write and graph the equation of a line from two points or from a slope and a point, and find its x- and y-intercepts
• Read the center and radius from a circle equation (x − h)² + (y − k)² = r², completing the square when it is expanded, and read the vertex from a parabola in vertex form y = a(x − h)² + k
• Finish on one ACT-hard question and say what made it hard

Coordinate geometry connects algebra and geometry on the xy-plane. Slope is rise over run: m = (y₂ − y₁) / (x₂ − x₁). In slope-intercept form y = mx + b, m is the slope and b is the y-intercept; point-slope form is y − y₁ = m(x − x₁). To graph a line, plot the y-intercept and use the slope to find a second point; the x-intercept is where y = 0. The distance between points (x₁,y₁) and (x₂,y₂) is √((x₂−x₁)² + (y₂−y₁)²). The midpoint is ((x₁+x₂)/2, (y₁+y₂)/2). Distinct nonvertical parallel lines have equal slopes; two vertical lines are also parallel. Nonvertical perpendicular lines have slopes that are negative reciprocals (e.g., 2 and −1/2). A horizontal line (slope 0) is perpendicular to a vertical line (undefined slope), so do not try to take a reciprocal of zero. A line in standard form Ax + By = C has slope −A/B (solve for y to see it): 3x − 9y = 4 has slope 1/3, not 3, so compare two standard-form lines only after finding each slope. Two lines meet where their equations are both true: write each line as y = mx + b (from two intercepts, the slope is −(y-intercept)/(x-intercept)), set the right sides equal, solve for x, then find y. The ACT frequently asks you to find the equation of a line given two points, or to determine the relationship between two lines. Circles and parabolas: (x − h)² + (y − k)² = r² has center (h, k) and radius r, so (x + 2)² + (y − 5)² = 49 has center (−2, 5) and radius 7, not 49. When the equation is expanded, complete the square in x and in y: x² + y² − 4x + 6y − 12 = 0 becomes (x − 2)² + (y + 3)² = 25, so the center is (2, −3) and the radius is 5. A parabola in vertex form y = a(x − h)² + k has vertex (h, k) and opens up when a > 0 and down when a < 0. To decide which equation matches a graph or a table, test two or three points in each choice rather than judging by shape, because a curve can look straight between two plotted points. Ellipses and transformations are in the Circles and Parabolas lesson. Coordinate geometry is a reliably tested chunk of the Math section, so these formulas earn back their memorization cost. The hardest questions, at the end of the section: chained questions, such as a line through a given point perpendicular to another line, then its intercepts, then the area of the triangle it forms with the axes. The traps are using the same slope instead of the negative reciprocal, dropping the 1/2 in the triangle area, and using a slanted side as the height.

• When a problem gives two points, find the slope first: m = (y₂ − y₁) / (x₂ − x₁).
• Memorize the distance and midpoint formulas. They appear on nearly every ACT.
• Slopes of horizontal lines are 0; slopes of vertical lines are undefined.
• If a question asks for the line perpendicular to y = 3x + 1, the slope is −1/3.