GEOMETRY
COORDINATE
GEOMETRY
Proofs
Name __________________________________
Period _________________________________
Table of Contents
Day 1: SWBAT: Use Coordinate Geometry to Prove Right Triangles and Parallelograms
Pgs: 2 – 8
HW: Pgs: 9 – 12
Day 2: SWBAT: Use Coordinate Geometry to Prove Rectangles, Rhombi, and Squares
Pgs: 13 - 18
HW: Pgs: 19 – 21
Day 3: SWBAT: Use Coordinate Geometry to Prove Trapezoids
Pgs: 22 - 26
HW: Pgs: 27 – 28
Day 4: SWBAT: Practice Writing Coordinate Geometry Proofs (REVIEW)
Pgs: 29 - 31
Day 5: TEST
Coordinate Geometry Proofs
Slope: We use slope to show parallel lines and perpendicular lines.
Parallel Lines have the same slope
Perpendicular Lines have slopes that are negative reciprocals of each other.
Midpoint: We use midpoint to show that lines bisect each other.
Lines With the same midpoint bisect each other
Midpoint Formula:
mid   x1  x2 , y1  y 2 
2 
 2
Distance: We use distance to show line segments are equal.
You can use the Pythagorean Theorem or the formula:
d  (x) 2  (y) 2
1
Day 1 – Using Coordinate Geometry To Prove Right Triangles and
Parallelograms
Warm – Up
Explaination:_______________________________________________________
__________________________________________________________________
Explaination:_______________________________________________________
__________________________________________________________________
2
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.
a 2 + b2 = c 2
(
)2 + (
)2 = (
)2
Statement:
3
“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.
 _____  _____
a 2 + b2 = c 2
(
)2 + (
)2 = (
)2
Statement:
4
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 ___________________________________________.
5
PRACTICE SECTION:
Example 1: Prove that the polygon with coordinates A(1, 1), B(4, 5), and C(4, 1) is a right
triangle.
Example 2: Prove that the polygon with coordinates A(4, -1), B(5, 6), and C(1, 3) is an
isosceles right triangle.
6
Example 3: Prove that the quadrilateral with the coordinates P(1,1), Q(2,4), R(5,6) and S(4,3)
is a parallelogram.
Challenge
7
SUMMARY
Proving Right Triangles
Proving Parallelograms
Exit Ticket
8
Homework
1.
2.
9
3.
4.
10
6. Prove that quadrilateral LEAP with the vertices
L(-3,1), E(2,6), A(9,5) and P(4,0) is a parallelogram.
11
7.
8.
9.
12
Day 2 – Using Coordinate Geometry to Prove Rectangles, Rhombi, and Squares
Warm – Up
1.
2.
13
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 ___________________________________________.
14
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 ___________________________________________.
15
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 _______________________________________________.
16
SUMMARY
Proving Rectangles
Proving Rhombi
Proving Squares
17
Challenge
Exit Ticket
1.
2.
18
DAY 2 - Homework
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 PLUS with the vertices P(2,1), L(6,3), U(5,5), and
S(1,3) is a rectangle.
19
3. Prove that quadrilateral DAVE with the vertices D(2,1), A(6,-2), V(10,1), and E(6,4) is a
rhombus.
4. Prove that quadrilateral GHIJ with the vertices G(-2,2), H(3,4), I(8,2), and
J(3,0) is a rhombus.
20
5. Prove that a quadrilateral with vertices J(2,-1), K(-1,-4), L(-4,-1) and M(-1, 2) is a square.
6. Prove that ABCD is a square if A(1,3), B(2,0), C(5,1) and D(4,4).
21
Day 3 – Using Coordinate Geometry to Prove Trapezoids
Warm - Up
1.
2.
22
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 ___________________________________________.
23
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 ___________________________________________.
24
Practice
Prove that the quadrilateral with the vertices C(-3,-5), R(5,1), U(2,3) and D(-2,0) is a trapezoid
but not an isosceles trapezoid.
25
Challenge
SUMMARY
Exit Ticket
26
Day 3 - Homework
1.
2.
27
3. .
4. Triangle TOY has coordinates T(-4, 2), O(-2, -2), and Y(2, 0). Prove TOY is an isosceles
right triangle.
28
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
29
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.
30
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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