Geometric Proofs using Coordinate Systems

How do we use coordinate geometry to prove that the diagonals of a parallelogram bisect each other?

The basic tools of coordinate geometry are the distance and midpoint formulas and equations of lines, all of which were discussed in previous lessons.

We can put any parallelogram ABCD in a plane, such that A is the origin and B is a point on the positive x-axis, i.e. A=(0,0) and B=(r,0). Let C=(u,v) and D=(s,t).  is parallel to , so the points C and D will have the same y-coordinate, i.e., v=t.
Furthermore, since  , u=r + s . So, in fact, C=(r + s, t).

The diagonals are the line segments  and . If P denotes the point of their intersection, we want to show that P is the midpoint of both the segments  and . We can find P by finding the equations of the lines containing  and .

The slope m₁ of the line L₁ containing  is . Recall that (y – y₁) = m(x – x₁) is the equation of the line with slope m containing the point (x₁, y₁). Because L₁ contains the origin, the equation of L₂ is . Similarly, the slope m₂ of the line L₂ containing BD is . Because L₂ contains B = (r, 0), the equation of L₂ is  .

To find x, set  and solve for x.

We can substitute x in either of the equations in order to find y.

So the coordinates of P are . If we apply the midpoint formula to A=(0,0) and C=(r+s, t), we get M_AC=  = P. Similarly, if we apply it to B=(r, 0) and D=(s, t), we get M_BD=  =P.

How do we prove that the diagonals of a rhombus are perpendicular?

Since a rhombus is a parallelogram, we can use exactly the same set-up that we used in the previous proof.

We need to show that diagonals  and  are perpendicular using the equations of the lines L₁ and L₂ containing them.

Review of New Vocabulary and Concepts

• Pythagorean Theorem : The square of the hypotenuse is equal to the sum of the squares of the legs in any right triangle.
• Converse of the Pythagorean Theorem : if the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle.
• Given a point P in the plane and let r be a positive real number . The circle with center P and radius r is the set of all points in the plane with distance r from P.
  
• A chord of a circle is a line segment with endpoints on the circle .
• The diameter of a circle is length of a chord containing the center of the circle and is denoted by d. The diameter of a circle is twice its radius.
• An arc is any connected part of a circle.
• Any two distinct points A and B of the circle divide it into two arcs called the minor arc AB and the major arc AXB.
• Arc Addition Theorem :If B is a point of the arc AC, then m() = m() + m().
  
• The measure of an inscribed angle is equal to half the measure of its intercepted arc.
• If two inscribed angles intercept the same arc or congruent arcs, then the angles are congruent.
  
• Two arcs in the same circle or congruent circles are congruent if and only if their corresponding chords are congruent.
• A secant of a circle is any line intersecting the circle at two points.
• A tangent of a circle is any line that intersects the circle at exactly one point called the point of tangency.
• A line is tangent to a circle if and only if the radius drawn to the point of tangency is perpendicular to the line.
• The measure of an angle formed by two secants that intersect in the interior of a circle is one half the sum of the arcs intercepted by the angle and its vertical angle.
• Arc length formula: s = ar, where a is the arc measure and r is the radius of the circle.
• The equation of a circle in the plane with center (h, k) and with radius r is .
  
• The standard form of the equation of a circle is x² + y² + Ax + By + C = 0.

Geometry (Ray Jurgenson and Richard G. Brown): Houghton Mifflin, 1999.

Geometry for Dummies (Wendy Arnone): Wiley Publishing, Inc., 2001.

Schaum’s Outline of Geometry (Barnett Rich, et. al.): McGraw Hill, 1999.

Standard Deviants: Geometry DVD 2-Pack (Flavia Colonna and Rebecca Berg): Cerebellum Corporation, 2000.

Don’t forget to test your knowledge with the Geometry and Spatial Reasoning Chapter Quiz;