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NY Regents Geometry study materials
NY Regents Geometry study resources

NY Regents Geometry: Expressing Geometric Properties with Equations

NY Regents Geometry · Work through your exam, one topic at a time.

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Expressing Geometric Properties with Equations

Expressing Geometric Properties with Equations

This domain tests coordinate geometry: distance, midpoint, slope, equations of lines and circles, and classifying figures from coordinates.

Core rules

  • Distance: d equals the square root of (x2 minus x1) squared plus (y2 minus y1) squared. Midpoint: the average of the x-coordinates and the average of the y-coordinates.
  • Slope: m equals (y2 minus y1) divided by (x2 minus x1). Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals, so their product is negative 1.
  • Point-slope form: y minus y1 equals m times (x minus x1). Horizontal lines have slope 0; vertical lines have undefined slope.
  • Circle with center (h, k) and radius r: (x minus h) squared plus (y minus k) squared equals r squared. Complete the square to convert general form to center-radius form.
  • To classify a quadrilateral, check both pairs of opposite sides, diagonals, and slopes. A parallelogram has diagonals that bisect each other; a rectangle has congruent diagonals; a rhombus has perpendicular diagonals; a square has both.

Worked example

A and B are endpoints of a diameter: A is (-2, 1) and B is (4, 5). Find the circle's equation. The center is the midpoint: ((-2 + 4)/2, (1 + 5)/2) equals (1, 3). The radius is half AB. AB equals the square root of (4 minus -2) squared plus (5 minus 1) squared, which is the square root of 36 plus 16, or the square root of 52, or 2 root 13, so r equals root 13. Thus the equation is (x minus 1) squared plus (y minus 3) squared equals 13. Check: r squared is 13, not 2 root 13.

Traps

Subtract coordinates in the same order; reversing one pair flips the slope sign. Do not set the radius equal to r squared in the circle equation. When completing the square, add the same value to both sides. A figure with four congruent sides is a rhombus, not necessarily a square; you still need a right angle. Vertical lines cannot be written in slope-intercept form, and a negative reciprocal is not the same as an opposite.