Answers for 6-2C

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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.

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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?

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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

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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

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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