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Geometric Proofs using Coordinate Systems

Objective

In this lesson, you will use coordinate systems to prove geometric theorems.

Previously Covered:

  • The diagonals of a parallelogram bisect each other.
  • The diagonals of a rhombus are perpendicular.

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 is the origin and 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 and will have the same y-coordinate, i.e., v=t.
Furthermore, since  ,
u=r + s . So, in fact, C=(r + s, t).

 Parallelogram ABCD in a plane

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

The slope mof the line Lcontaining  is Recall that (y – y1= m(x – x1) is the equation of the line with slope containing the point (x1, y1). Because Lcontains the origin, the equation of Lis Similarly, the slope mof the line Lcontaining BD is . Because Lcontains B = (r, 0), the equation of Lis  .

To find x, set  and solve for x.

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

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

 Rhombus

We need to show that diagonals  and  are perpendicular using the equations of the lines Land Lcontaining them.

Question

What step must we show to finish the proof?

  1. Show that the midpoint of  is the midpoint of .
  2. Show that the slopes of Land Lare reciprocals.
  3. Show that the slopes of Land Lare negative reciprocals.
  4. Show that  ~.

Reveal Answer

The correct answer is C . Two perpendicular lines have slopes that are negative reciprocals.

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 be a positive real number . The circle with center and radius r is the set of all points in the plane with distance 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 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 (hk) and with radius 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;

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