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Geometry Section 6-2D Quadrilaterals and Coordinate proofs Page 433 Be ready to grade 6-2C Quiz Thursday. Answers for 6-2C. Rectangle Rhombus Parallelogram Square x = 30, y = 60(2 pts.) x = 8, y = 22(2 pts.) x = 8(1 pt.) Not possible Drawing – a rectangle No –(2 pts.) - PowerPoint PPT Presentation
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Geometry Section 6-2D Geometry Section 6-2D Quadrilaterals and Quadrilaterals and Coordinate proofsCoordinate proofs
Page 433Page 433Be ready to grade 6-2CBe ready to grade 6-2CQuiz ThursdayQuiz Thursday
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Answers for 6-2C1. Rectangle2. Rhombus3. Parallelogram4. Square5. x = 30, y = 60 (2 pts.)6. x = 8, y = 22 (2 pts.)7. x = 8 (1 pt.)8. Not possible9. Drawing – a rectangle10. No – (2 pts.)11. Diagonals bisect each other12. Diagonals bisect and are perpendicular13. Sketch with no lines of symmetry14. Sketch with 2 lines of symmetry15. Sketch with 4 lines of symmetry
18 pts. possible
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GRADE SCALE – 18 POSSIBLE
17.5 – 97%17 – 94%16.5 – 92%16 – 89%15.5 – 86%15 – 83%14.5 – 81%14 – 78%13.5 – 75%13 – 72%12.5 – 69%12 – 67%
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Coordinate Geometry:Coordinate Geometry:
Many proof can be made easier using coordinate geometry. To use this method, we first place the
figure on a coordinate plane so that one vertex is at the origin and one side is on an axis.
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Coordinate Geometry:Coordinate Geometry:
Review: How can you calculate the slope of a line on a coordinate plane?
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Slope =riserun
y2 –y1
x2 – x1
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Coordinate Geometry:Coordinate Geometry:
Review: What is true about the slopes of perpendicular lines?
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The slopes of the 2 lines will be
negative reciprocals of each other.
If we want to prove that sides of a
quadrilateral are perpendicular, we will prove their
slopes multiplied equal -1.
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Coordinate Geometry:Coordinate Geometry:
Review: What is true about the slopes of parallel lines?
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The slopes of the 2 lines will be
identical.
If we want to prove that sides of a
quadrilateral are parallel, we will
prove their slopes are the same.
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Coordinate Geometry:Coordinate Geometry:If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the
coordinates of the other points.
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If E is at (0,0) and we know that F is 9 units
away, what are the coordinates of F?
E F
(9,0)
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Coordinate Geometry:Coordinate Geometry:If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the
coordinates of the other points.
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If E is at (0,0) and we know that F is 18 units away, what are the coordinates of F?
(18,0)
E F
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Coordinate Geometry:Coordinate Geometry:If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the
coordinates of the other points.
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If E is at (0,0) and we know that F is x
units away, what are the coordinates
of F?
(x,0)
E F
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Coordinate Geometry:Coordinate Geometry:If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the
coordinates of the other points.
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If E is at (0,0) and we know that F is a
units away, what are the coordinates
of F?E F
(a,0)
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Coordinate Geometry:Coordinate Geometry:If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the
coordinates of the other points.
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If E is at (0,0) and we know that G is 9
units away, what are the coordinates
of G?E
G
(0,9)
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Coordinate Geometry:Coordinate Geometry:If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the
coordinates of the other points.
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If A is at (0,0) and we know that B is b
units away, what are the coordinates
of B?A
B
(0,b)
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Coordinate Geometry:Coordinate Geometry:If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the
coordinates of the other points.
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If I wanted to move B 3 places to the right, what would it’s coordinate be?
A
B
(0+3,b)
(3,b)
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Coordinate Geometry:Coordinate Geometry:If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the
coordinates of the other points.
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If I wanted to move B 4 places to the
left, what would it’s coordinate be?
A
B
(0-4,b)
(-4,b)
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Coordinate Geometry:Coordinate Geometry:If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the
coordinates of the other points.
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If I wanted to move B x places to the
left, what would it’s coordinate be?
A
B
(0-x,b)
(-x,b)
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Coordinate Geometry:Coordinate Geometry:If I know that one point of a shape will always be on (0,0) then we can use variables to indicate the
coordinates of the other points.
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If we know no specific numbers
except the (0,0), we use variables and
equations to give the other coordinates.E
G
We’ll work across the bottom first.
E = (0,0)
F
F = (a,0)G = (b,c)
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Coordinate Geometry:Coordinate Geometry:To find point D, we take the labeled height and an
equation to show the shift left or right.
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G
E = (0,0)F F = (a,0)
G = (b,c)
H
H = (a+b,c)
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Example:Example:Place a parallelogram on the coordinate plane. Whenever possible, place vertices on the axes.
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Let two vertices be the origin and (a,0).
Place a vertex at (b,c).
The last vertex insures that opposite sides have the same
slope. Choose (a+b,c) to make another horizontal side
and a second side of slope c/b.
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Explore:Explore:To use a coordinate proof, we will not use the 2 column proof. We will use the paragraph form.
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Place square ABCD, with side length a on a coordinate plane.
Label the vertices and give their coordinates.
A –B –C –D –
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Explore:Explore:To use a coordinate proof, we will not use the 2 column proof. We will use the paragraph form.
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Use the slope formula to find the slopes of the diagonals
AC and BD.
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Explore:Explore:To use a coordinate proof, we will not use the 2 column proof. We will use the paragraph form.
Pg.435
Use your results from the previous screen to show that
AC and BD are perpendicular. Explain.
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Reflect:Reflect:What properties can be used to show that lines are parallel or perpendicular when doing
a coordinate proof?
Pg.435
If they are parallel, slopes will be equal. If they are
perpendicular, slopes will be negative reciprocals.
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Reflect:Reflect:Why is the use of coordinates
a helpful strategy in some proofs?
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Coordinates can be used to calculate lengths and slopes.
We can use that information to prove congruence, parallel and
perpendicular.
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Reflect:Reflect:In trapezoid MNPQ, you
could give point P the coordinates (c,d). However, there is a better choice for
them. What are they? Explain.
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(a,c) is better because the distance on the x axis is the
same for pts. N and P.
Hint: don’t choose a new variable if there’s any way to use the other variables. Sometimes, the problem will tell you
not to introduce a new variable. In that case, make an equation of some kind.
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Exercises:Exercises:The figure shown is an
isosceles trapezoid. What are the coordinates of vertex C?
(Hint: You will have to introduce one new variable.) What are the coordinates of
vertex D?
#1
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C is (c,b)
D is (c+a,0)
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Exercises:Exercises:Assign coordinates to the vertices. Do not introduce
any new variables.
#2
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(b,a)
ABCD is a rectangle.
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Exercises:Exercises:Assign coordinates to the vertices. Do not introduce
any new variables.
#3
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(a+c,b)
DEFG is a parallelogram.
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Exercises:Exercises:Assign coordinates to the vertices. Do not introduce
any new variables.
#4
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(a, a3)
HIJ is equilateral.
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Exercises:Exercises:Use the figure to verify each
statement.
#5
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The opposite sides of FGHJ are parallel.
Slope of HJ =
Slope of GF =
Slope of JF =
Slope of HG =
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Exercises:Exercises:Use the figure to verify each
statement.
#6
Pg.436
The opposite sides of FGHJ are congruent.
JF =
HG =
HJ =
GF =
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Exercises:Exercises:Use a coordinate proof to prove the following:
If a quadrilateral is a square, then its diagonals are congruent.
#8
Pg.436
The distance from (0,0) to (a,a) is a2+a2 = a2
The distance from (0,a) to (a,0) is a2+(-a)2 = a2
The diagonals are congruent.
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Exercises:Exercises:Use a coordinate proof to prove the following:
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
#9
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The midpoint of DB is
The midpoint of AC is
The midpoints are identical, so the diagonals bisect.
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Homework: Practice 6-2DQuiz Thursday