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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:
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POLYGONS
Quadrilaterals
Parallelograms
RECTANGLES Has ALL the properties of the shapes above it
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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
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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.
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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
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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.
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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
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Method 2: Use the Distance Formula,
to determine whether opposite sides are congruent.
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Since each pair of opposite sides of the quadrilateral have the same measure, they are congruent. Quadrilateral ABCD is a parallelogram.
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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
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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.
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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
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If you wanted to install a ceiling fan in your rectangular family room,Find where it would be in the room?
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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?
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homework