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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.
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:
Given: 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:
The slope formula is used to determine the slope of DE.
The slope of AB would be:
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:
Given: rectangle ABCD with points: A (0,0), B (0,b), C (d,b) D (d,0); AC and BD are diagonals.
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.
The right triangle above is placed on the coordinate plane with x and y coordinates indicated.
The coordinates of the midpoint of the hypotenuse also correspond to the midpoint of the distance between each vertex of the sides.
The distance equation for line segment AD is:
The distance equation for line segment BD is:
The distance equation for line segment CD is:
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
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.
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.
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