Return to Free Math Homework Help   >>   Return to Geometry Main Page

Geometry: Coordinate Proofs





 



Set Theory
Points, Lines, Planes
Angles
Transversals
Planes and Space
Triangles
Quadrilaterals
Polygons
Circles
Tangents and Secants
Chords, Power Point
Proofs
Congruence
Congruent Triangles
Congruent Polygons
Congruency Proofs
Congruent Circles, Arcs
Similarity
Similarity Proofs
Pythagorean Theorem
Symmetry and Transformations
Analytic Geometry
Coordinate Proofs
Solids
Non-Euclidean Geometry
Rules Postulates Theorems
Glossary
Morleys Theorem
Fractals






In analytic geometry, proofs are done using the coordinate plane. A geometric figure is plotted and certain rules and formulas are used to prove the hypothesis. Some of these formulas include the Pythagorean Theorem in the form of the distance formula, the midpoint formula, slope formulas and circles. Algebra is one of many forms of math used to arrive at the conclusion.

It is helpful, sometimes, to include a diagram located in the coordinate plane. When placing the figure it is a good idea to follow certain guidelines.

  1. Use variables in the form of letters or use zero to represent coordinate values.
  2. Set a vertex or the center of a polygon or the center or edge of a circle at the origin.
  3. Have the drawing satisfy the requirements of the hypothesis without unnecessary detail.
  4. Try to locate the geometric figure along one or more axes of the coordinate plane.
  5. For simpler calculations use positive coordinates by placing the figure in Quadrant I of the plane.
  6. The use of symmetry can simplify a problem.
  7. Line the figure up horizontally or vertically if there are parallel or perpendicular components.
  8. Use as few variables as possible.
A geometric figure on the coordinate plane should be lined up in a way that is easiest to prove the hypothesis.

Also, it can help sometimes to work in reverse from the proof in order to understand the logical flow from the beginning. The following proof involves a triangle on the coordinate plane that is divided by a line segment. It is used to prove the following theorem:

Triangle Midsegment
The midsegment of a triangle is parallel to the third side.

A triangle's midsegment is parallel to the side it doesn't connect to.

Given: triangle.ABC with midsegment DE and with vertex points: top C is (0,c), bottom left A is (0,0), bottom right B is (b,0)

Prove: DE || AB

Points D and E are midpoints with the following coordinates:

D = (0+0
2
, 0+c
2
) = (0 , c
2
)

E = (0+b
2
, c+0
2
) = (b
2
, c
2
)

The slope formula is used to determine the slope of DE.

m   =   y2 y1
x2 - x1

mDE   =   
c
2
- c
2

b
2
- 0


mDE   =   0
b
2
   =   0

The slope of AB would be:

mAB   =   0 0
b - 0
   =   0

The slopes are equal, mAB = mDE so the line segments are parallel, DE || AB.



Rectangle Diagonals Proof

The next proof involves the diagonals of a rectangle. The applicable theorem can be stated as follows:

Rectangle Diagonals
The diagonals of a rectangle are equal.


Given: rectangle ABCD with points: A (0,0), B (0,b), C (d,b) D (d,0); AC and BD are diagonals.

A rectangle in the coordinate plane is used to prove that its diagonals are equal.

Prove: AC = BD

The rectangle with one of its diagonals creates two right triangles. As a result, the hypotenuse of these triangles can be calculated and compared since they represent the diagonals of the rectangle.

AC = √(d 0)2 + (b - 0)2 = √d2 + b2

BD = √(d - 0)2 + (0 - b)2 = √d2 + b2

Since both AC and BD are equal to √d2 + b2, AC = BD by substitution.



Hypotenuse Midpoint Proof

The Pythagorean Theorem and its other form, the distance formula, is used often in coordinate geometry proofs. The next proof involves the midpoint of the hypotenuse. It is based on the following theorem.

Hypotenuse Midpoint
In a right triangle, the midpoint of a hypotenuse is equidistant from the vertices.


The midpoint of the hypotenuse of a right triangle is the same distance away from all the vertices.

The right triangle above is placed on the coordinate plane with x and y coordinates indicated.

To prove that the hypotenuse midpoint is equidistant to the vertices, the right triangle is placed in the coordinate plane.

The coordinates of the midpoint of the hypotenuse also correspond to the midpoint of the distance between each vertex of the sides.

Let the vertices of a right triangle be represented by: A (0,0), B (b,0), C (0,c) with hypotense midpoint
(b
2
,c
2
).

The distance equation for line segment AD is:

The first of three equations calculating the distance from the hypotenuse midpoint to a vertex in the right triangle.

The distance equation for line segment BD is:

The second of three equations calculating the distance from the hypotenuse midpoint to a vertex in the right triangle.

The distance equation for line segment CD is:

The third of three equations calculating the distance from the hypotenuse midpoint to a vertex in the right triangle.

Since the definition of a midpoint says that the bisected line segment has equal parts, the following line segment lengths are equal, BD = CD. Also, since line segments AD and BD have the same result from the distance formula, they are also equal, AD = BD. Therefore, the following statement holds, AD = BD = CD. As a result, the vertices are all an equal distance from the hypotenuse midpoint. Geometry formulas are frequently used in coordinate proofs. The next proof uses the midpoint formula on the diagonals of a parallelogram.



Bisected Parallelogram Diagonals

Bisecting Parallelogram Diagonals
The diagonals of a parallelogram bisect each other.


To prove that the diagonals of a parallelogram bisect each other, the parallelogram is placed in a coordinate plane.

Let a parallelogram be defined in the coordinate plane at the following vertices: A (-a,0), B (0,b), C (2a, b), D (a,0); line segments AC and BD are diagonals of the parallelogram.

The following formulas calculate the location of the midpoint for each diagonal.

MAC   =   (-a+2a
2
, 0+b
2
)   =   (a
2
,b
2
)

MBD   =   (0+a
2
, b+0
2
)   =   (a
2
,b
2
)

As a result, MAC = MBD, the midpoints are the same and with the definition of the midpoint the diagonals of the parallelogram bisect each other. Sometimes, proofs are written in a paragraph format as shown in the previous proofs instead of using the two column method.

 




<<    Previous
Next    >>
Analytic Geometry
Solids





Return to Free Math Homework Help   >>   Return to Geometry Main Page