Name ________________________
Worksheet 2.2
Coordinate Proofs
Geometric ideas/theorems can be proved utilizing algebraic coordinate techniques. The key is to
apply a convenient coordinate system to the geometric object. The following problem below
illustrates how this can be done:
Proving a Theorem using Coordinate Geometry
Here is a theorem regarding one of the properties of
parallelograms, which we will now prove:
S (b, c)
R
The diagonals of a parallelogram bisect each other.
Step 1: Place the desired figure
P (0, 0)
Q (a, 0)
Here is a clever way to place a parallelogram PQRS
on a coordinate plane! What advantages do we have
given this placement given this particular figure?
Step 2: Label the figure
Three of the vertices are P(0, 0), Q(a, 0) and S(b, c).
What are the coordinates of vertex R?
Step 3: Prove the desired statement and state the conclusion
Find the coordinates of the midpoint of PR and the midpoint of SQ . They should be the
same point, which allows us to state our conclusion.
IMPORTANT: To be effective and efficient with coordinate proofs, choosing the most convenient
place to put your shape on the axis is important, as is making sure you don’t over define your
shape
(i.e. don’t make a triangle a right triangle unless you have been given this to be the case). In
addition, you should be able to recognize what shapes are made by certain points on the Cartesian
Coordinate Plane.
Exercises:
1. Prove that the diagonals of a rectangle are congruent.
2. Prove that the medians bisecting the congruent sides of an isosceles triangle are congruent.
Hint: think of a clever location on the
coordinate axes for this isosceles triangle,
especially given what you are trying to prove!
1. Use a coordinate proof to show that the opposite sides of a parallelogram are congruent.
2. Use a coordinate proof to show that an isosceles trapezoid has congruent diagonals. Hint: think of a
clever location on the coordinate axes for this isosceles trapezoid, given what you are trying to
prove!
3. Prove that the diagonals of a rhombus are perpendicular.
Hint: think of a clever location on the coordinate axes for this rhombus, given what you are
trying to prove – be careful not to over-define your rhombus!
4. Two altitudes are drawn to the legs of an isosceles triangle. Prove that these altitudes have slopes
of opposite signs.
5.
Prove that the midsegment of a trapezoid (the segment connecting the midpoints of the two nonparallel sides) is parallel to the bases and has length equal to the average of the lengths of the bases.
6.
Draw parallelogram ABCD and connect the midpoints.
a. On the grid below, label the midpoints with coordinates based on b,c, and d.
y
D (c, d)
A (0,0)
C (b+c, d)
B (b, 0)
x
b. Conjecture: the resulting “midpoint” quadrilateral is a: ____________________
c. Prove your result
(NOTE: you might also need to prove what the quadrilateral is not.)
7.
A triangle is formed by the intersection points of the lines:
y  17 ( x  3) , y  3( x  3) , and y   x  5
a. Graph these lines and find
the coordinates for the
vertices of the triangle
analytically.
b. Find the area of the triangle.
8.
Analytically show that the point (4, 4) is on the perpendicular bisector of the line segment
joining (−2, 12) and (4, −6).
9. Given quadrilateral ABCD, with A(-5, -2), B(-3, 4), C(3, 6), and D(1, 0):
a. Show ABCD is a parallelogram
(choose your method).
6
5
4
3
2
1
8
6
4
2
2
1
2
3
4
5
6
b. Given that ABCD is a parallelogram, now verify ABCD is a rhombus by
i. Showing one pair of adjacent sides are congruent
ii. Showing that the diagonals are perpendicular
4
6
8