Unit 6: Connecting Algebra and Geometry through Coordinates

Proving Coordinates of Rectangles and Squares

Characteristics of Rectangles and Squares

(both are parallelograms)

Rectangles:• Opposite sides are

parallel and congruent• Opposite angles are

congruent and consecutive angles are supplementary

• All four angles are right angles (90°)

• Diagonals bisect each other and are congruent

Squares:• All sides are congruent• Opposite sides are parallel• All four angles are right

angles• Diagonals bisect each other

and are congruent• Diagonals are perpendicular• Diagonals bisect opposite

angles

Using the Distance Formula

We will be using the distance formula to prove that given coordinates form a square or a rectangle. Remember the distance formula is derived from the Pythagorean Theorem:

Also recall that:• Parallel lines have the same slope• Perpendicular lines have slopes that are negative

(opposite sign) reciprocals whose product is -1.

Example 1: Prove that the following vertices represent the vertices of a rectangle.

P (3, 1) Q (3, -3) R (-2, -3) S (-2, 1)

Practice proving that vertices represent particular geometric figures by using all possible characteristics.

Use the chart for quadrilaterals to help you remember the properties.

S P

QR

Example 1: Prove that the following vertices represent the vertices of a rectangle.

P (3, 1) Q (3, -3) R (-2, -3) S (-2, 1)

Property: Opposite sides are congruent.

Find the length of each side:

= 4 ; = 4

= 5 ; = 5

S P

QR

Example 1: Prove that the following vertices represent the vertices of a rectangle.

P (3, 1) Q (3, -3) R (-2, -3) S (-2, 1)

Property: Opposite sides are parallel.

Find the slopes of opposite sides:

m = 0 ; m = 0

m = undefined; m = undefined

S P

QR

Example 1: Prove that the following vertices represent the vertices of a rectangle.

P (3, 1) Q (3, -3) R (-2, -3) S (-2, 1)

Property: Angles are 90ᵒ.

Find the slope of adjacent sides:

m = 0 ; m = undefined

m = undefined; m = 0

S P

QR

Example 1: Prove that the following vertices represent the vertices of a rectangle.

P (3, 1) Q (3, -3) R (-2, -3) S (-2, 1)

Property: Angles are 90ᵒ.

Find the slope of adjacent sides:

m = undefined ; m = 0 ;

m = undefined; m = 0

S P

QR

Example 1: Prove that the following vertices represent the vertices of a rectangle.

P (3, 1) Q (3, -3) R (-2, -3) S (-2, 1)

Property: Diagonals are congruent.

Find the length of each diagonal:

=

=

S P

QR

Example 1: Prove that the following vertices represent the vertices of a rectangle.

P (3, 1) Q (3, -3) R (-2, -3) S (-2, 1)

Property: Diagonals bisect each other. Find the midpoints of each diagonal:

Midpoint = =

Midpoint = =

Diagonals bisect each other (have the same mid-point).

S P

QR

Example 1: Prove that the following vertices represent the vertices of a rectangle.

P (3, 1) Q (3, -3) R (-2, -3) S (-2, 1)

PQRS is a rectangle because:

and ; Opposite sides are congruent.

and ; Opposite sides are parallel.

, , , ; All angles are 90ᵒ.

; Diagonals are congruent.

Midpoint = Midpoint ; Diagonals bisect each other.

Example 2: Prove that the following vertices represent the vertices of a square.

P (5, 2) Q (2, 5) R (-1, 2) S (2, -1)

S

P

Q

R

Practice proving that vertices represent particular geometric figures by using all possible characteristics.

Use the chart for quadrilaterals to help you remember the properties.

Example 2: Prove that the following vertices represent the vertices of a square.

P (5, 2) Q (2, 5) R (-1, 2) S (2, -1)

S

P

Q

R

Property: All sides are congruent.

Find the length of each side: =

=

=

=

Example 2: Prove that the following vertices represent the vertices of a square.

P (5, 2) Q (2, 5) R (-1, 2) S (2, -1)

S

P

Q

R

Property: All sides are congruent.

Find the length of each side: =

=

=

=

Example 2: Prove that the following vertices represent the vertices of a square.

P (5, 2) Q (2, 5) R (-1, 2) S (2, -1)

S

P

Q

R

Property: All sides are congruent

Find the length of each side:

= ; = ;

= ; =

Example 2: Prove that the following vertices represent the vertices of a square.

P (5, 2) Q (2, 5) R (-1, 2) S (2, -1)

S

P

Q

R

Property: Opposite sides are parallel.

Find the slope of each side:

m of = = = 1

m of = = = 1

Example 2: Prove that the following vertices represent the vertices of a square.

P (5, 2) Q (2, 5) R (-1, 2) S (2, -1)

S

P

Q

R

Property: Opposite sides are parallel.

Find the slope of each side:

m of = = = -1

m of = = = -1

Example 2: Prove that the following vertices represent the vertices of a square.

P (5, 2) Q (2, 5) R (-1, 2) S (2, -1)

S

P

Q

R

Property: All angles are 90ᵒ.

Use the slopes of each side:

m of = 1 ; m of = -1

m of = -1 ; m of = 1

m of = 1 ; m of = -1

m of = -1 ; m of = 1

Example 2: Prove that the following vertices represent the vertices of a square.

P (5, 2) Q (2, 5) R (-1, 2) S (2, -1)

S

P

Q

R

Property: Diagonals are congruent.

Find the length of each diagonal: =

6

=

6

Example 2: Prove that the following vertices represent the vertices of a square.

P (5, 2) Q (2, 5) R (-1, 2) S (2, -1)

S

P

Q

R

Property: Diagonals bisect each other.

Find the mid-point of each diagonal: Midpoint = =

Midpoint = =

Diagonals bisect each other (have the same mid-point).

Example 2: Prove that the following vertices represent the vertices of a square.

P (5, 2) Q (2, 5) R (-1, 2) S (2, -1)

S

P

Q

R

Property: Diagonals are perpendicular.

Find the slopes of each diagonal:

m of =

m of =

Diagonals are perpendicular.

Example 2: Prove that the following vertices represent the vertices of a square.

P (5, 2) Q (2, 5) R (-1, 2) S (2, -1)

PQRS is a square because:

; All sides are congruent.

and ; Opposite sides are parallel.

, , , ; All angles are 90ᵒ.

; Diagonals are congruent.

Midpoint = Midpoint ; Diagonals bisect each other.

; Diagonals are perpendicular.

Example 3: Do the given vertices represent those of a rectangle? Why or why not?

P (5, 2) Q (1, 9) R (−3, 2) S (1, −5)

S

P

Q

R

Check for congruency of opposite sides.

= = = = Since all sides are congruent, this could be a rectangle (more specifically, a square). Now check slopes for 90ᵒ angles.

What would be the most obvious why to begin if you do not have a diagram?

Example 3: Do the given vertices represent those of a rectangle? Why or why not?

P (5, 2) Q (1, 9) R (−3, 2) S (1, −5)

S

P

Q

R

Check slopes for possible 90ᵒ angles.

m of = = =

m of = = = −

Since adjacent angles are opposite signs only (and not reciprocals), these vertices do not represent those of a rectangle.What kind of quadrilateral is PQRS?

Summary of the Proof Process• When you are told that vertices are those of a certain quadrilateral,

you may assume that the properties of that quadrilateral are present.• When you are simply told vertices, often you must determine if those

vertices represent a specific type of quadrilateral.• Begin with an easy property to rule out possible types, such as length.• Proceed with each additional and required property to verify a type of

quadrilateral.• For all HW, you must first state which property you are testing and

show all work to support your conclusions.• If you have a graph, you may count vertical or horizontal units to

determine length, otherwise you must use the distance formula.• If you have vertical or horizontal segments, you may write undefined

or 0 for the slope. Otherwise, you must use the slope formula.• Clearly state your conclusions in a complete sentence.