2.3c: Quadrilaterals

M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems involving polygons

GSE’s

- Rectangle

G-GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).

G-CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

CCSS:

POLYGONS

Quadrilaterals

Parallelograms

RECTANGLES Has ALL the properties of the shapes above it

In Addition to all those:

Rectangles also have:A

1) Congruent Diagonals

2) All 4 angles are right

What can we do to prove these on the coordinate plane?

BR

N

Use Distance Formula for AB and RN

Use slope to find if each consecutive side is opposite reciprocals

Quadrilateral RSTU is a rectangle. If and find x.

The diagonals of a rectangle are congruent,

Definition of congruent segments

Substitution

Subtract 6x from each side.

Add 4 to each side.

Quadrilateral LMNP is a rectangle. Find x and y.

908523 xx

90108 x808 x10x

322)10(3

588)10(5

NLPm

MLNm

5y

306y

3226y

AnglesInterior Alternate

PLNMNL

58 32

Quadrilateral ABCD has vertices A(–2, 1), B(4, 3), C(5, 0), and D(–1, –2). Determine whether ABCD is a rectangle using the Slope Formula.

Method 1: Use the Slope Formula, to see if

consecutive sides are perpendicular.

Answer: The perpendicular segments create four right angles. Therefore, by definition ABCD is a rectangle.

quadrilateral ABCD is a parallelogram. The product of the slopes of consecutive sides is –1. This means that

Method 2: Use the Distance Formula,

to determine whether opposite sides are congruent.

Since each pair of opposite sides of the quadrilateral have the same measure, they are congruent. Quadrilateral ABCD is a parallelogram.

The length of each diagonal is

Answer: Since the diagonals are congruent, ABCD is a rectangle.

Find the length of the diagonals.

If we know the shape is a Parallelogram, we can determine if it is a rectangle by the lengths of the diagonals

Quadrilateral WXYZ has vertices W(–2, 1), X(–1, 3), Y(3, 1), and Z(2, –1). Determine whether WXYZ is a rectangle using the Distance Formula.

Determine whether each statement is True or False. Explain

• If the quadrilateral is a rectangle, then it is a parallelogram

• If the quadrilateral is a parallelogram, then it is a rectangle.

• If a parallelogram has congruent diagonals, then it is a rectangle

• If a quadrilateral has congruent diagonals, then it is a rectangle

True, all rectangles are parallelograms

False, not all parallelograms are rectangles, they could be other shapes as well.

True, by definition, a parallelogram with congruent diagonals is a rectangle

False, a 4 sided figure (quadrilateral) could have congruent diagonals and not be a rectangle

If you wanted to install a ceiling fan in your rectangular family room,Find where it would be in the room?

The foundation for your house was poured, and you measured the outside walls to Be 40’ by 24’ . Does this make it rectangular? Why or why not?

homework