Regents Examination in Geometry: Coordinate Geometry Proofs

TOOLS AND STRATEGIES OF COORDINATE GEOMETRY PROOFS

Key Ideas
Special properties of triangles, quadrilaterals, and other figures can be proven using coordinate geometry. Certain tools are used in coordinate geometry:
Distance—to show segments congruent
Midpoint—to show a segment is bisected
Slope—to show that segments and lines are parallel (same slope) or perpendicular (negative reciprocal slopes)


Tools of Coordinate Geometry Proofs
In a coordinate geometry proof, the coordinates of the vertices of a figure are provided as givens, and we are asked to prove some feature of the figure using coordinate geometry. The coordinate geometry tools at our disposal are distance, midpoint, and slope.

Equal distances indicate congruence. Slope can be used to demonstrate parallel or perpendicular lines and segments. Midpoints show if a segment is bisected.
 

Quantity	Formula	Use
Slope	or	Prove segments or lines are parallel (slopes are equal) or perpendicular (slopes are negative reciprocals)
Midpoint		Prove segments bisect each other (midpoints are concurrent)
Length		Segments are congruent (distances between endpoints are equal)

Follow these steps when writing a coordinate geometry proof:
Graph the points. The graph will help you plan a strategy, check your work, and help with some calculations.
Plan a strategy. What property will you demonstrate, and what calculations are needed?
Perform the calculations. Write down the general equations you are using, and clearly label which segments correspond to which calculations.
Write a summary statement. You must state in words a justification that explains why your calculations justify the proof.

PARALLELOGRAM PROOFS


Key Ideas
The midpoint, distance, and slope formulas can be used to prove quadrilaterals are parallelograms, rectangles, rhombi, or squares.

The following tools are used:
Midpoint can show that diagonals bisect each other.
Distance can show sides or diagonals are congruent.
Slope can show sides are parallel, sides are perpendicular, or diagonals are perpendicular.


Proving Parallelograms
A given quadrilateral can be proven to be a parallelogram by demonstrating one of the parallelogram properties. The three properties that can be easily proven are parallel sides, congruent sides, and diagonals bisecting each other.

The calculations required and suggested summary statements are shown in the following table:

When writing a coordinate geometry proof, be sure to show all formulas used. Show all substitutions and calculations with labels to make it clear which parts are being used in the proof. You must conclude with a summary paragraph that states what was proven and how it was done. You can think of the summary as a short paragraph proof.

Example
The coordinates of quadrilateral ABCD are A(−2, 1), B(4, 4), C(6, 5), and D(0, 2). Prove ABCD is a parallelogram.
Solution:
Strategy—find the midpoint of diagonals  and  to show the diagonals bisect each other.

Midpoint of :



The midpoint of  is (2, 3)


Midpoint of :

The midpoint of  is (2, 3)

Summary—The midpoints of  and  are the same, so the diagonals bisect each other. ABCD is a parallelogram because a quadrilateral whose diagonals bisect each other is a parallelogram.

Proving Special Parallelograms
To prove a quadrilateral is a special parallelogram, first prove the figure is a parallelogram. Then demonstrate one of the special properties shown in the table that follows.

For squares, you need to show that the figure is both a rectangle and a rhombus.

The method you pick depends on what calculations you are most comfortable with. Midpoint and slope calculations are usually faster to do than distance calculations. However, finding the distance may make more sense if you need to follow up by showing the figure is a special parallelogram.

Some suggested strategies are:
Parallelogram—midpoints of the diagonals
Rectangle—slope of each side
Rhombus—length of each side or midpoint and the slope of the diagonals

Example 1
Given coordinates A(0, 2), B(4, 8), C(7, 6), and D(3, 0), prove ABCD is a rectangle.
Solution:
Strategy—Calculate the slopes of the four sides.


The slopes of opposite sides are equal, so ABCD is a parallelogram. The slopes of consecutive sides are negative reciprocals, making the consecutive sides perpendicular and the angles right angles. ABCD is a rectangle because a parallelogram with right angles is a rectangle.

Example 2
Given coordinates R(−3, 0), S(1, 7), T(9, 6), and U(5, −1), prove RSTU is a rhombus.

Solution:
Strategy—Calculate the midpoints and slopes of the diagonals.

The diagonals of RSTU have the same midpoint, therefore they bisect each other, making RSTU a parallelogram. The diagonals of RSTU have negative reciprocal slopes, making them perpendicular. RSTU is a rhombus because a parallelogram with perpendicular diagonals is a rhombus.

Example 3
Given coordinates M(−4, 2), A(0, 5), T(3, 1), and H(−1, −2), prove MATH is a square.

Solution:
Strategy—Calculate the midpoints, slopes, and lengths of the diagonals.

The midpoints of the diagonals are the same.

So the diagonals bisect each other, making RSTU a parallelogram. The slopes of the diagonals are negative reciprocals, making them perpendicular. RSTU is a rhombus because it is a parallelogram with perpendicular diagonals. The lengths of the diagonals are equal, making them congruent. RSTU is a rectangle because it is a parallelogram with congruent diagonals. RSTU is a square because it is both a rectangle and a rhombus.

TRIANGLE PROOFS

Key Ideas
Coordinate geometry tools can be used to show the following:

1. A triangle is scalene, isosceles, or equilateral.
2. A triangle is a right triangle.
3. Segments in triangles are altitudes or medians.
4. Figures are congruent.

Triangles
Prove a triangle is scalene, isosceles, or equilateral by finding the length of each side. Equal lengths indicate congruent sides.

Example 1
△TBX has vertices with coordinates T(1, 2), B(5, 10), and X(9, 2). Prove △TBX is isosceles but not equilateral.
Solution:
Strategy—Find the length of each side, and show that two sides are congruent.


TB = BX, but they are not equal to TX. Therefore △TBX has two congruent sides and is isosceles but not equilateral.
Use slope to determine if sides are perpendicular to prove a triangle is a right triangle.

Example 2
Given points D(−2, 0), J(2, 3), and R(−1, 7), prove △DJR is a right triangle and identify which side is the hypotenuse.
Solution:
Strategy—Find the slope of each side.

and  are negative reciprocals, so they are perpendicular. ∠J is therefore a right angle, and △DJR is a right triangle. The side opposite ∠J is the hypotenuse, so  is the hypotenuse.

Coordinate geometry can be used to prove special segments in triangles.

The following table summarizes some of the segments and points of concurrency and also the calculations needed.

Example 3
The vertices of △QRS are Q(0, 9), R(8, 5), and S(3, 0). Point T(2, 3) lies on . Prove  is an altitude.

Solution:

 and  have negative reciprocal slopes, so they are perpendicular.
Therefore  is an altitude of △QRS.

Example 4
△RST has vertices with coordinates R(1, 1), S(7, 3), and T(2, 7). If point V lies on  and has coordinates (4, 2), prove  is a median.
Solution:

Since V is the midpoint of ,  is a median.

If you need to find the centroid of a triangle, you can take either a graphical or an algebraic approach. You can graph the three medians and identify the point of concurrency, or you can find the point algebraically that divides the median in a 1 : 2 ratio.

Example 5
The vertices of △ABC are A(0, 0), B(2, 6), and C(10, 6). Find the centroid of △ABC graphically. Then demonstrate algebraically that the centroid divides a median in a 1 : 2 ratio.
Solution: Let D, E, and F be the midpoints of , , and , respectively.

The coordinates of the midpoints are found with the midpoint formula.

Graph the medians to locate the centroid. The centroid is located at G(4, 4).

Use the distance formula to confirm the ratio.

Point G does divide the segment in a 1 : 2 ratio.
Alternatively, find the point that divides the median into a 1 : 2 ratio and show its coordinates are (4, 4).

Example 6
Prove algebraically that D(2, 3) is the circumcenter of △ABC, whose vertices have coordinates A(2, −2), B(−1, 7), and C(5, 7).

Solution: D will be the circumcenter if it is equidistant from points A, B, and C.

Use the distance formula
.

D is equidistant from each of the vertices of the triangle, so it must be the circumcenter of the triangle.