Sign in to save

Bookmark this page so you can find it later.

Sign in to save

Bookmark this page so you can find it later.

Coordinate Proofs Using the Distance and Midpoint Formulas cheat sheet - grade 9-11

Click image to open full size

Math Grade 9-11

Coordinate Proofs Using the Distance and Midpoint Formulas Cheat Sheet

A printable reference covering distance, midpoint, slope, congruent segments, bisected diagonals, parallel lines, and perpendicular lines for grades 9-11.

TopicsAlgebraGraphsShapesMeasurementGeometryStudy Guide
Download PNG

Related Tools

Related Labs

Related Worksheets

Related Infographics

Study as Flashcards

Coordinate proofs use algebra to prove geometric facts on the coordinate plane. This cheat sheet helps students organize the distance formula, midpoint formula, and slope formula into clear proof strategies. These tools are useful for proving that segments are congruent, diagonals bisect each other, sides are parallel, and angles are right.

A strong coordinate proof shows both the calculation and the geometric conclusion it supports.

The most important formulas are the distance formula d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}, the midpoint formula M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right), and the slope formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. Equal distances prove congruent segments, equal midpoints prove segments bisect each other, equal slopes prove parallel lines, and slopes with product −1-1 prove perpendicular lines. These facts are often combined to prove quadrilaterals are parallelograms, rectangles, rhombi, or squares.

Labeling points carefully and matching each calculation to a reason makes the proof clear.

Key Facts

  • The distance between A(x1,y1)A(x_1,y_1) and B(x2,y2)B(x_2,y_2) is AB=(x2−x1)2+(y2−y1)2AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
  • The midpoint of A(x1,y1)A(x_1,y_1) and B(x2,y2)B(x_2,y_2) is M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right).
  • The slope of a nonvertical line through A(x1,y1)A(x_1,y_1) and B(x2,y2)B(x_2,y_2) is m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
  • Two segments are congruent if their distances are equal, so AB=CDAB = CD means AB‾≅CD‾\overline{AB} \cong \overline{CD}.
  • Two segments bisect each other if they have the same midpoint, so MAC=MBDM_{AC} = M_{BD} proves the diagonals share a midpoint.
  • Two nonvertical lines are parallel if their slopes are equal, so m1=m2m_1 = m_2 proves the lines are parallel.
  • Two nonvertical lines are perpendicular if their slopes are negative reciprocals, so m1m2=−1m_1m_2 = -1 proves a right angle.
  • A quadrilateral is a parallelogram if both pairs of opposite sides are parallel or if its diagonals bisect each other.

Vocabulary

Coordinate proof
A proof that uses coordinates, formulas, and algebra to show that a geometric statement is true.
Distance formula
A formula, d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}, used to find the length between two points.
Midpoint formula
A formula, M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right), used to find the point halfway between two endpoints.
Slope
The ratio m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} that measures the steepness and direction of a line.
Parallel lines
Lines in the same plane that do not intersect and have equal slopes when both are nonvertical.
Perpendicular lines
Lines that intersect to form right angles and have slopes whose product is −1-1 when both slopes are defined.

Common Mistakes to Avoid

  • Mixing up the order of coordinates in the slope formula is wrong because xx and yy differences must match the same point order, such as m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
  • Forgetting the square root in the distance formula is wrong because (x2−x1)2+(y2−y1)2(x_2 - x_1)^2 + (y_2 - y_1)^2 gives the squared distance, not the actual length.
  • Claiming lines are perpendicular just because the slopes have opposite signs is wrong because perpendicular nonvertical slopes must be negative reciprocals with m1m2=−1m_1m_2 = -1.
  • Using equal side lengths alone to prove a quadrilateral is a parallelogram is incomplete unless both pairs of opposite sides are shown congruent or another valid parallelogram test is used.
  • Assuming diagonals bisect each other without calculating both midpoints is wrong because a coordinate proof must show MAC=MBDM_{AC} = M_{BD} with matching coordinates.

Practice Questions

  1. 1 Find the distance between A(−2,3)A(-2,3) and B(4,−5)B(4,-5) using d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
  2. 2 Find the midpoint of C(7,−1)C(7,-1) and D(−3,5)D(-3,5) using M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right).
  3. 3 For A(0,0)A(0,0), B(6,2)B(6,2), C(8,8)C(8,8), and D(2,6)D(2,6), use slopes to determine whether ABCDABCD is a parallelogram.
  4. 4 Explain why showing that two diagonals have the same midpoint is enough to prove a quadrilateral is a parallelogram.