Geometry Pointers

• Distance Formula


  If P is a point (x, y) and Q is another point (x2, y2), then the distance from P to Q is: √((x-x2)2+(y-y2)2)
• Midpoint Formula


  If P is a point (x, y) and Q is another point (x2, y2), then the midpoint M(x, y) of line segment PQ is: ((x+x2)/2, (y+y2)/2)
• Slope

  The slope (m) of the line passing through points (x1, y1) and (x2, y2) is given by the formula: m = (y2-y1)/(x2-x1)
• A line with a positive slope rises from left to right. A line with a negative slope falls from left to right
• The slope of a horizontal line is zero. The slope of a vertical line is undefined
• Slope-intercept form


  A linear equation in the form: y = mx + b, where m is the slope and b is the y-intercept
• Point-slope form


  A linear equation in the form: y-y1 = m(x-x1), where (x1, y1) is a point on the line and m is the slope
• Lines having the same slope are parallel unless when they are collinear. Conversely, parallel lines have the same slope
• If two non-vertical lines are perpendicular, their slopes are negative reciprocals of each other
• Vertical angles


  • The pairs of angles which are not adjacent to each other when two straight noncollinear lines intersect
  • The measures of the vertical angles are equal
• Congruent angles


  Two angles whose measures are equal
• Parallel lines cut by a transversal


  • The alternate interior angles are congruent
  • The corresponding angles are congruent
  • The interior angles on the same side of the transversal are supplementary
• Right angle


  An angle whose value is 90°, with the two sides perpendicular to each other
• Straight angle


  An angle whose measure is 180°, with the sides forming a straight line
• Acute angle


  An angle less than 90°
• Complementary angles

  Two angles whose measurements sum up to 90°
• Supplementary angles

  Two angles whose measurements sum up to 180°
• The three angles of any triangle add up to 180°
• An exterior of a triangle is equal to the sum of the remote interior angles
• Similar triangles

  Triangles that have the same shape, with corresponding sides proportional
• Triangle Inequality Theorem

  The length of one side of a triangle must be greater than the difference and less than the sum of the other two sides
• Isosceles Triangle

  A triangle that has two equal sides, with the angles opposite the equal sides (base angles) also equal
• Equilateral Triangle


  A triangle in which all three sides are equal, and all three angles measure 60°. The altitude equals the side times √3
• Pythagorean Triples


  Sets of numbers that satisfy the Pythagorean Theorem, where the square of the hypotenuse equals the sum of the squares of the other two sides
• 30°-60°-90° Triangle

  The leg opposite the 30° angle equals half the hypotenuse, the leg opposite the 60° angle equals half the hypotenuse times √3, and the ratio of the shorter leg to the hypotenuse is 1:2
• 45°-45°-90° Triangle

  The hypotenuse equals a leg times √2, and the leg equals half the hypotenuse times √2
• The sum of the interior angles of a regular polygon with n sides is (n-2) x 180°
• The sum of the exterior angles of a regular polygon with n sides is 360°, and each exterior angle measures 360°/n
• Parallelogram

  • Opposite sides are parallel and congruent
  • Opposite angles are congruent
  • Consecutive angles are supplementary
  • Diagonals bisect each other
  • Each diagonal bisects the parallelogram into two congruent triangles
• Rectangle

  • All angles are right angles
  • Diagonals are congruent
• Rhombus

  • All sides are congruent
  • Diagonals are perpendicular
  • Diagonals bisect the angles
• A square is a rectangular rhombus, so it has all the properties of a rectangle and rhombus
• Surface Area of a Rectangular Solid

  SA = 2LW + 2WH + 2LH, where L is length, W is width, and H is height
• Surface Area of a Cube

  SA = 6s^2, where s is the side length