answers for 6-2c
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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 PresentationTRANSCRIPT
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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.
Pg.433
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Coordinate Geometry:Coordinate Geometry:
Review: How can you calculate the slope of a line on a coordinate plane?
Pg.433
Slope =riserun
y2 –y1
x2 – x1
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Coordinate Geometry:Coordinate Geometry:
Review: What is true about the slopes of perpendicular lines?
Pg.433
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?
Pg.433
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.
Pg.433
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.
Pg.433
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.
Pg.433
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.
Pg.433
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.
Pg.433
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.
Pg.433
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.
Pg.433
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.
Pg.433
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.
Pg.433
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.
Pg.433
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.
Pg.433 E
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.
Pg.433
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.
Pg.435
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.
Pg.435
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?
Pg.435
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.
Pg.435
(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
Pg.435
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
Pg.436
(b,a)
ABCD is a rectangle.
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Exercises:Exercises:Assign coordinates to the vertices. Do not introduce
any new variables.
#3
Pg.436
(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
Pg.436
(a, a3)
HIJ is equilateral.
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Exercises:Exercises:Use the figure to verify each
statement.
#5
Pg.436
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
Pg.436
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