Geometry
Geometry in Regents Algebra I is largely coordinate geometry and transformations. The exam tests slope, parallel and perpendicular lines, distance and midpoint, directed segments, and how transformations affect figures. You may be asked to justify a geometric relationship algebraically or to write an equation from given points. Formulas appear on the reference sheet, but choosing the right one and keeping signs straight is the real work.
Core rules
- Slope is (y2−y1)/(x2−x1). Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals, so m1·m2=−1.
- The midpoint of a segment is the average of the coordinates: ((x1+x2)/2,(y1+y2)/2). The distance formula comes from the Pythagorean theorem.
- A point that divides a directed segment in a given ratio can be found by weighting the endpoints according to the ratio, not by using the midpoint unless the ratio is 1:1.
- Rigid motions-translations, rotations, reflections-preserve distance and angle measure, so they produce congruent figures. Dilations preserve shape but change size; the scale factor controls lengths, and area changes by the square of the scale factor.
- Vertical lines have undefined slope; horizontal lines have slope 0.
Worked example
Write the equation of the line through (2,−1) perpendicular to y=(2/3)x+4. The given slope is 2/3, so the perpendicular slope is −3/2. Use point-slope form: y−(−1)=−3/2(x−2), or y+1=−3/2(x−2). Simplified, y=−3/2x+2. The reasoning is to take the negative reciprocal first, then substitute the point.
Traps
Do not use the reciprocal without changing the sign for a perpendicular slope. Do not reverse the order of subtraction in slope or distance and then drop a sign. Do not assume a figure is congruent after a dilation. Check whether an equation should be vertical or horizontal before forcing it into y=mx+b.