GEOMETRY
COORDINATE
GEOMETRY
Proofs
Name __________________________________
Period _________________________________
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Coordinate Proof Help Page
Formulas
Slope:
Distance:
√
To show segments are congruent: Use the distance formula to find the length of the sides and
show that the sides have the same lengths and are therefore congruent.
To show that lines are perpendicular: Find the slopes of the lines and show that their products is -1.
To show that two lines are parallel: Find the slopes of the lines and show that they are equal.
Right Triangle
Method 1: Show that the Pythagorean Theorem works.
Method 2: Show that two of the sides are perpendicular.
Isosceles Triangle
Show that 2 sides are congruent.
Equilateral Triangle
Show that all three sides are congruent.
Rhombus
Show that all sides are congruent.
Parallelogram
Method 1: Show that both pairs opposite sides are parallel.
Method 2: Show that both pairs of opposite sides are congruent.
Rectangle
Method 1: Show that both pairs of opposite sides are parallel and there is a right angle.
Method 2: Show that both pairs of opposite sides are congruent and diagonals are congruent.
Square
Method 1: Show that all four sides are congruent and there is a right angle.
Method 2: Show that all four sides are congruent and diagonals are congruent.
Trapezoid
Show that one pair of sides are parallel and the other are not.
Isosceles Trapezoid
Show that the polygon is a trapezoid (see above) and show that the nonparallel sides are
congruent.
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Table of Contents
Day 1
Essential Question: How can I use coordinate geometry to
prove right triangles and parallelograms
In class: Pages 5-7
Homework: Page 8
Day 2
Essential Question: How can I use coordinate geometry to
prove rectangles, rhombi, and squares.
In class: Pages 9-11
Homework: Page 12
Day 3
Essential Question: How can I use coordinate geometry to
prove trapezoids.
In class: Pages 13-14
Homework: Page 15
Day 4:
Students will practice writing coordinate geometry proofs
(review)
In class: Pages 16-18
Homework: Finish anything that has not yet been completed.
3
Warm-Ups
Day 1:
Graph AB with coordinates A(-2, -5) and B(6,-1).
Graph CD with coordinates C(-1,3) and D(2,-3).
Find the slope of AB and CD
Are the lines parallel or perpendicular?
Day 2:
1. QUVX is a rectangle. Find the value of x and y.
2. Find the value of a, b, and c in the rhombus.
Day 3:
4
Proving a triangle is a right triangle
Method: Calculate the distances of all three sides and then test the Pythagorean’s theorem to
show the three lengths make the Pythagorean’s theorem true.
“How to Prove Right Triangles”
1. Prove that A (0, 1), B (3, 4), C (5, 2) is a right triangle.
Show:
Formula:
d  (x) 2  (y) 2
Work: Calculate the Distances of all three sides to show Pythagorean’s Theorem is true.
a2 + b2 = c2
(
)2 + (
)2 = (
)2
Statement:
5
“How to Prove an Isosceles Right Triangles”
Method: Calculate the distances of all three sides first, next show two of the three sides are
congruent, and then test the Pythagorean’s theorem to show the three lengths make the
Pythagorean’s theorem true.
2. Prove that A (-2, -2), B (5, -1), C (1, 2) is a
an isosceles right triangle.
Show:
Formula:
d  (x) 2  (y) 2
Work: Calculate the Distances of all three sides to show Pythagorean’s Theorem is true.
 _____  _____
a2 + b2 = c2
(
)2 + (
)2 = (
)2
Statement:
6
Proving a Quadrilateral is a Parallelogram
Method: Show both pairs of opposite sides are equal by calculating the distances of all four sides.
Examples
3. Prove that the quadrilateral with the coordinates L(-2,3), M(4,3), N(2,-2) and O(-4,-2) is a
parallelogram.
Show:
Formula:
d  (x) 2  (y) 2
Work: Calculate the Distances of all four sides to show that the opposite sides are equal.
 _____  _____
AND
_____  _____
Statement:
 ________ is a parallelogram because ___________________________________________.
7
Homework- Day 1
1.
2.
3.
8
Proving a Quadrilateral is a Rectangle
Method: First, prove the quadrilateral is a parallelogram, then that the diagonals are congruent.
Examples:
1. Prove a quadrilateral with vertices G(1,1), H(5,3), I(4,5) and J(0,3) is a rectangle.
Show:
Formula:
d  (x) 2  (y) 2
Work
Step 1: Calculate the Distances of all four sides to show
that the opposite sides are equal.
 _____  _____
AND
_____  _____
 ________ is a parallelogram because ___________________________________________.
Step 2: Calculate the Distances of both diagonals to show they are equal.
 _____  _____
Statement:
 __________ is a rectangle because ___________________________________________.
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Proving a Quadrilateral is a Rhombus
Method: Prove that all four sides are congruent.
Examples:
2. Prove that a quadrilateral with the vertices A(-1,3), B(3,6), C(8,6) and D(4,3) is a rhombus.
Show:
Formula:
d  (x) 2  (y) 2
Work
Step 1: Calculate the Distances of all four sides to show
that all four sides equal.
 _____  _____

_____  _____
Statement:
 __________ is a Rhombus because ___________________________________________.
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Proving that a Quadrilateral is a Square
Method: First, prove the quadrilateral is a rhombus by showing all four sides is congruent; then
prove the quadrilateral is a rectangle by showing the diagonals is congruent.
Examples:
3. Prove that the quadrilateral with vertices A(-1,0), B(3,3), C(6,-1) and D(2,-4) is a square.
Show:
Formula:
d  (x) 2  (y) 2
Work
Step 1: Calculate the Distances of all four sides to show
all sides are equal.
 _____  _____

_____  _____
 ________ is a ____________.
Step 2: Calculate the Distances of both diagonals to show they are equal.
 _____  _____
 __________ is a _______________.
Statement:
 __________ is a Square because _______________________________________________.
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Homework- Day 2
1. Prove that quadrilateral ABCD with the vertices A(2,1),
B(1,3), C(-5,0), and D(-4,-2) is a rectangle.
2. Prove that quadrilateral DAVE with the vertices D(2,1), A(6,2), V(10,1), and E(6,4) is a rhombus.
3. Prove that ABCD is a square if A(1,3), B(2,0), C(5,1) and
D(4,4).
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Proving a Quadrilateral is a Trapezoid
Method: Show one pair of sides are parallel (same slope) and one pair of sides are not
parallel (different slopes).
Example 1: Prove that KATE a trapezoid with coordinates K(0,4), A(3,6), T(6,2) and E(0,-2).
Show:
Formula:
Work
Calculate the Slopes of all four sides to show
2 sides are parallel and 2 sides are nonparallel.
 _____
_____
and
_____
_____
Statement:
 __________ is a Trapezoid because ___________________________________________.
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Proving a Quadrilateral is an Isosceles Trapezoid
Method: First, show one pair of sides are parallel (same slope) and one pair of sides are
not parallel (different slopes). Next, show that the legs of the trapezoid are congruent.
Example 2: Prove that quadrilateral MILK with the vertices M(1,3), I(-1,1), L(-1, -2), and
K(4,3) is an isosceles trapezoid.
Show:
Formula:
d  (x) 2  (y) 2
Work
Step 1: Calculate the Slopes of all four sides to show
2 sides are parallel and 2 sides are nonparallel.
Step 2: Calculate the distance of
both non-parallel sides
(legs) to show legs congruent.
Statement:
 __________ is an Isosceles Trapezoid because ___________________________________________.
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Homework- Day 3
1.
2.
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Day 4 – Practice writing Coordinate Geometry Proofs
1. The vertices of ABC are A(3,-3), B(5,3) and C(1,1). Prove by coordinate geometry that
ABC is an isosceles right triangle.
2. Given ABC with vertices A(-4,2), B(4,4) and C(2,-6), the midpoints of AB and BC are P
and Q, respectively, and PQ is drawn. Prove by coordinate geometry:
a. PQ║ AC
b. PQ = ½ AC
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3. Quadrilateral ABCD has vertices A(-6,3), B(-3,6), C(9,6) and D(-5,-8). Prove that
quadrilateral ABCD is:
c. a trapezoid
d. not an isosceles trapezoid
4. The vertices of quadrilateral ABCD are A(-3,-1), B(6,2), C(5,5) and D(-4,2). Prove that
quadrilateral ABCD is a rectangle.
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5. The vertices of quadrilateral ABCD are A(-3,1), B(1,4), C(4,0) and D(0,-3). Prove that
quadrilateral ABCD is a square.
6. Quadrilateral METS has vertices M(-5, -2), E(-5,3), T(4,6) and S(7,2). Prove by coordinate
geometry that quadrilateral METS is an isosceles trapezoid.
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