Download Coordinate Geometry

Coordinate Geometry
Formulas
y 2  y1
 Distance Formula: d 
x2  x1
Parallel Lines  equal slopes
Perpendicular Lines  slopes are negative reciprocals
Perpendicular Lines  product of the slopes is equal to -1
 Slope =
x2  x1 2   y2  y1 2
 x  x1 y 2  y1 
 Midpoint =  2
,

2 
 2
Summary of Methods for Coordinate Geometry Proofs
To prove that a figure is isosceles or equilateral, use:
1. The distance formula to show equal lengths.
To prove that a triangle is a right triangle, use:
1. The distance formula to verify the Pythagorean Theorem; or
2. The slope formula to show that two sides of the triangle are perpendicular.
To prove that a quadrilateral is a parallelogram, use:
1. The distance formula to show that both pairs of opposite sides are congruent; or
2. The midpoint formula to show that the diagonals bisect each other; or
3. The slope formula to show that both pairs of opposite sides are parallel; or
4. The slope formula to show that one pair of opposite sides is parallel, and then the distance formula to
show the same sides are congruent.
To prove that a parallelogram is a rectangle, use:
1. The distance formula to show that the diagonals of the parallelogram are congruent; or
2. The slope formula to show that two consecutive sides of the parallelogram are perpendicular.
To prove that a parallelogram is a rhombus, use:
1. The distance formula to show that two consecutive sides of the parallelogram are congruent; or
2. The slope formula to show that the diagonals of the parallelogram are perpendicular.
To prove that a rectangle is a square, use:
1. The distance formula to show that consecutive sides are congruent; or
2. The slope formula to show that diagonals of the rectangle are perpendicular.
To prove that a rhombus is a square, use:
1. The distance formula to show that the diagonals are congruent; or
2. The slope formula to show that two consecutive sides of the rhombus are perpendicular.
To prove that a quadrilateral is a trapezoid, use:
1. The slope formula to show that one pair of opposite sides are parallel and that the other pair of
opposite sides are not parallel.
To prove that a quadrilateral is a kite, use:
1. The slope formula to show that the diagonals are perpendicular; AND
2. The distance formula to show that two sides are not congruent.
Properties of Triangles and Quadrilaterals
Triangles (by sides)
Scalene Triangle – Triangle with three equal sides
Isosceles Triangle – Triangle with two equal sides (legs) and two congruent angles (base angles)
Equilateral Triangle – Triangle with three congruent sides and three congruent angles (60o)
Triangles (by angles)
Acute Triangle – Triangle with three acute angles
Obtuse Triangle – Triangle with one obtuse angle
Right Triangle – Triangle with one right angle
Equiangular Triangle – Triangle with three congruent angles and three congruent sides
Quadrilaterals
Parallelogram
1. Diagonals divide the parallelogram into two congruent triangles.
2. Opposite sides of a parallelogram are congruent.
3. Opposite sides of a parallelogram are parallel.
4. Two consecutive angles of a parallelogram are supplementary.
5. The diagonals of a parallelogram bisect each other.
Rectangle
1. All properties of a parallelogram.
2. Contains four right angles.
3. Diagonals are congruent.
Rhombus
1.
2.
3.
4.
All properties of a parallelogram.
Four sides are congruent. (equilateral)
Diagonals are perpendicular.
Diagonals bisect the vertices.
Square
1. All properties of a parallelogram.
2. All properties of a rhombus.
3. Four right angles.
Trapezoid
1. One pair of opposite sides are parallel.
Isosceles Trapezoid
1. One pair of opposite sides are parallel.
2. Two legs are congruent.
3. Diagonals are congruent.
4. Base angles, angles at the end of the bases, are congruent.
Kite:
1. Two consecutive sides are congruent.
2. Diagonals are perpendicular.
3. One pair of opposite angles are congruent.