Many proofs that take several steps to accomplish in a two column or paragraph proof can be much more easily proven using coordinate geometry. The results from earlier courses that we may need to use in our proofs are summarized below.
The slope of PQ is
The length of PQ is
The coordinates of M, the midpoint of PQ, are
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The equation of the line through P(x1,y1) with slope m is y - y1 = m(x - x1)
Slopes of perpendicular lines are negative reciprocals of each other. This means they have a product of -1. Parallel lines have equal slopes.
Example 1:
Use coordinate geometry to prove that the midpoint of the hypotenuse of a right triangle is equidistant from the vertices of the triangle.
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Remarks: The most important step in a coordinate geometry proof is placing the figure onto the coordinate plane in such a way as to make the mathematical manipulations as simple as possible. Clearly had we chosen the coordinates of the vertices of the right triangle as shown at the left, our calculations would have been much more involved. We would have had to use the fact that AC and BC have slopes that are negative reciprocals of each other somewhere in our proof because of the right angle at C. Not having any of the vertices at the origin or on an axis complicates our calculations as well. |
Example 2:
Prove that the median drawn to the base of an isosceles triangle is perpendicular to the base.
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Remark: It may have been tempting to choose the coordinates of the isosceles triangle as shown at left, but this immediately would assume that the median is perpendicular to the base since the axes are perpendicular. With this orientation, one would not use the given fact that the triangle is isosceles. If we were given that the median drawn to one side of a triangle was perpendicular to that side and were then asked to prove that the triangle was isosceles, the layout would be ideal. |
Assignment
For questions 1 to 6 determine the coordinates of the remaining points in each figure if the figure is to be:
| 1. a square | 2. a rectangle |
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| 3. a parallelogram | 4. any triangle |
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| 5. an isosceles trapezoid | 6. any quadrilateral |
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| 7. a rhombus | |
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Use coordinate geometry to prove each of the following statements. You can use the coordinates established in questions 1 to 7 to assist your diagram layout.
8. The diagonals of a rectangle are congruent.
9. The line segment joining the midpoints of two sides of a triangle is (a) parallel to the remaining side of the triangle and (b) is half as long as the remaining side.
10. The diagonals of a rhombus are perpendicular.
11. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
12. The diagonals of an isosceles trapezoid are congruent.
13. If the midpoint of one side of a triangle is equidistant from all three vertices, then the triangle is a right triangle.
14. The median of a trapezoid is parallel to the two bases and has a length equal to the average of the lengths of the bases.
15. The figure formed by joining the midpoints of consecutive sides of any quadrilateral is a parallelogram.
