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GEOMETRY HONORS

COORDINATE

GEOMETRY

Proofs

Name __________________________________

Period _________________________________

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Coordinate Geometry Proofs

Distance formula: 2 2

2 1 2 1( ) ( )d x x y y = √(∆𝑥)2 + (∆𝑦)2

Midpoint Formula: MP = 1 2 1 2( ) ( ),

2 2

x x y y

= (�̅�, �̅�)

Slope Formula: 2 1

2 1

y ym

x x

=

∆𝑦

∆𝑥

Recall: 2 lines are parallel if their slopes are =

2 lines are if their slopes are opposite reciprocals

Equation of a line: Slope Intercept form: y = mx + b

Point Slope Form: 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

Pythagorean’s Theorem: 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐

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Method for Coordinate Geometry Proofs:

To Prove a Polygon

is:

Prove the following Formulas used

Isosceles Triangle Find the distance of all 3 sides

Show 2 out of the 3 sides are congruent

Distance Formula

Right Triangle

(Use only one of these two

methods)

1. Calculate the Slope of all 3 sides

Show that a pair of sides are ⊥

(showing right angles are formed)

1. slope formula

2. Use Pythagorean’s Theorem

Show that if Pythagorean’s

theorem is true, then the

triangle must be a right triangle.

2. Pythagorean’s Theorem

Parallelogram

(Use only one of these four

methods)

1. both pairs of opposite sides are parallel 1. slope formula

2. both pairs of opposite sides are

congruent

2. distance formula

3. one pair of opposite sides are parallel and

congruent

3. slope and distance formula

4. diagonals bisect each other 4. midpoint formula

Rectangle

(Use only one of these two

methods)

1. Find the slope of all 4 sides.

First show it’s a parallelogram

because opposite sides are

parallel Then prove it’s a rectangle by

showing it’s a parallelogram

with right angles.

(Adjacent sides are ⊥)

1. Slope formula

2. Find the distance of all 4 sides and the

distance of both diagonals.

First show it’s a parallelogram

because opposite sides are

congruent Then prove it’s a rectangle by

showing it’s diagonals are

congruent.

2. distance formula

Rhombus Find the distance of all 4 sides

Show all sides are congruent (showing it’s a parallelogram and rhombus)

distance formula

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Square

(Use only one of these two

methods)

1. Find the distance of all 4 sides and both

diagonals. (6 – calculations)

First show all 4 sides are

congruent. (showing it’s a

parallelogram and rhombus)

Then show both diagonals are

congruent. (showing it’s a

rectangle)

1. distance formula

2. Find the slope of all 4 sides and both

diagonals. (6 – calculations)

First show it’s a parallelogram

because opposite sides are

parallel and then prove it’s a

rectangle by showing it’s a

parallelogram with right angles.

(Adjacent sides are ⊥)

Then prove it’s a rhombus by

showing it’s diagonals are ⊥

2. Slope formula

Trapezoid

Find the slope of all 4 sides

Show 1 pair of sides are parallel

and the other two sides are not parallel

Slope formula

Isosceles Trapezoid

(Use only one of these two

methods)

1. Find the slope of all 4 sides & the

distance of 2 non-parallel sides.

(6 – calculations)

First show 1 pair of sides are

parallel and the other two

sides are not parallel

Next, show the non-parallel

sides are congruent by using

the distance formula

Slope & Distance formula

2. Find the slope of all 4 sides & the

distance of both diagonals.

(6 – calculations)

First show 1 pair of sides are

parallel and the other two

sides are not parallel

Next, show the diagonals are

congruent by using the

distance formula.

Slope & Distance formula

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Day 1 – Using Coordinate Geometry To Prove Right Triangles and Parallelograms

Proving a triangle is a right triangle

Method 1: Show two sides of the triangle are perpendicular by demonstrating their slopes are opposite

reciprocals.

Method 2: 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.

Example 1: Prove that the triangle with coordinates A(4, -1), B(5, 6), and C(1, 3) is isosceles.

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Example 2: Prove that the polygon with coordinates A(1, 1), B(4, 5), and C(4, 1) is a right triangle.

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Proving a Quadrilateral is a Parallelogram

Example 3: Prove that the quadrilateral with the coordinates L(-2,3), M(4,3), N(2,-2) and O(-4,-2) is a

parallelogram.

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Proving a Quadrilateral is a Rectangle

Example 4: Prove a quadrilateral with vertices G(1,1), H(5,3), I(4,5) and J(0,3) is a rectangle.

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Proving a Quadrilateral is a Rhombus

Example 5: Prove that a quadrilateral with the vertices A(-2,3), B(2,6), C(7,6) and D(3,3) is a rhombus.

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Proving that a Quadrilateral is a Square

Example 6: Prove that the quadrilateral with vertices A(0,0), B(4,3), C(7,-1) and D(3,-4) is a square.

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Homework

1.

2.

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3.

4.

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5.

6. Prove that quadrilateral LEAP with the vertices L(-3,1), E(2,6), A(9,5) and P(4,0) is a parallelogram.

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7. kjhjh

1.

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5. Prove that quadrilateral ABCD with the vertices A(2,1), B(1,3), C(-5,0), and

D(-4,-2) is a rectangle.

6. Prove that quadrilateral PLUS with the vertices P(2,1), L(6,3), U(5,5), and

S(1,3) is a rectangle.

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10.Prove that quadrilateral DAVE with the vertices D(2,1), A(6,-2), V(10,1), and E(6,4) is a rhombus.

11.Prove that quadrilateral GHIJ with the vertices G(-2,2), H(3,4), I(8,2), and

J(3,0) is a rhombus.

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12.Prove that a quadrilateral with vertices J(2,-1), K(-1,-4), L(-4,-1) and M(-1, 2) is a square.

13.Prove that ABCD is a square if A(1,3), B(2,0), C(5,1) and D(4,4).

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Day 2 – Using Coordinate Geometry to Prove Trapezoids

Proving a Quadrilateral is a Trapezoid

Warm - Up

Determine the most specific name for quadrilateral ABCD with vertices,

A(2,-1), B (8,1), C (7,4) and D (1,2).

Example 1:

1. Prove that KATE a trapezoid with coordinates K(1,5), A(4,7), T(7,3) and E(1,-1).

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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.

3. 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.

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Homework

1.

2.

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3. fdf

4. Determine the perimeter of the ABC, A(-1,3) B(3,5) C (2,-4) (leave your answer in simplest radical form)

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6. Determine the most specific name for the quadrilateral ABCD with vertices,

A (0,0), B (3,4), C (0,8) and D (-3,4).

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Day 3 – Calculating the Areas of Polygons in the Coordinate Geometry

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Practice

1. Find the area of trapezoid ABCD if the vertices are A(1,5), B(7,3), C(2,-4) and D(-7,-1).

2. If the coordinates of the vertices of polygon PEACH are P(1,1), E(10,4), A(7,8), C(2,9) and H(-3,3), what is the area of pentagon

PEACH?

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3.

4. Given the lengths of the three sides of ABC, use Heron’s Formula to determine the area.

( )( )( )A s s a s b s c where a, b, and c are sides and 2

a b cs

. (round to two decimal places)

a) a = 3 cm, b = 5 cm , c = 6 cm b) a = 8 cm, b = 4 cm , c = 7 cm c) a = 6 cm, b = 10 cm , c = 8 cm

Area = ___________ cm2 Area = ___________ cm

2 Area = ___________ cm

2

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5. Find the area of a triangle whose vertices are (-5,4), (2,1) and (6,5).

Method 1: Box Technique

Method 2: Heron’s Formula

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Homework

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SUMMARY

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