attention teachers:
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Attention Teachers:• The main focus of section 7 should be proving that the
four coordinates of a quadrilateral form a _______. 9th and 10th graders should be shown the theoretical proofs (with a’s and b’s etc.) 11th and 12th grade teachers should focus on use of pythagorean/distance, slope, midpoint formulas, etc.
There is also a review of the area formulas for quadrilaterals. They should know triangle, rectangle, parallelogram, & square. They will be given the formulas for a trapezoid, kite, and rhombus on the test. They will need to connect the formula with the correct shape and use it.
• For homework I will provide you with two worksheets with appropriate problems (one for Thursday, one for Friday) 9th and 10th grade teachers add on coordinate proof exercises listed at the end of the lesson.
Lesson 6-7
Coordinate Proof with Quadrilaterals
Five-Minute Check (over Lesson 6-6)
Main Ideas
California Standards
Example 1: Positioning a Square
Example 2: Find Missing Coordinates
Example 3: Coordinate Proof
Example 4: Real-World Example: Properties ofQuadrilaterals
• Position and label quadrilaterals for use in coordinate proofs.
• Prove theorems using coordinate proofs.
Standard 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. (Key)Standard 17.0 Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles. (Key)
Parallelograms
RhombusRectangleSquare
Kites
Trapezoids
Isosceles Trapezoids
Quadrilaterals
Tree Diagram
Quadrilateral
Parallelogram Kite Trapezoid
Rhombus Rectangle
Square
Isosceles Trapezoid
Coordinate Proofs• Graph it!• What do you think it is?• Look for parallel lines (use slope formula.)• Look for congruent sides( use distance
formula.)– Congruent diagonals Rectangle or Iso.
Trapezoid.
Process for Positioning a Square
The y-coordinate of B is 0 because the vertex is on the x-axis. Since the side length is a, the x-coordinate is a.
1. Let A, B, C, and D be vertices of a rectangle with sides a units long, and sides b units
long.
Place the square with vertex A at the origin, along the
positive x-axis, and along the positive y-axis. Label the vertices A, B, C, and D.
2.
3.
Positioning a Square
Sample answer:
D is on the y-axis so the x-coordinate is 0. Since the side length is b, the y-coordinate is b.
The x-coordinate of C is also a. The y-coordinate is 0 + b or b because the side is b units long.
4.
5.
Position and label a square with sides a units long on the coordinate plane. Which diagram would best achieve this?
A. B.
C. D.
Find Missing Coordinates
Name the missing coordinates for the isosceles trapezoid.
Answer: D(b, c)
The legs of an isosceles trapezoid are congruent and have opposite slopes. Point C is c units up and b units to the left of B. So, point D is c units up and b units to the right of A. Therefore, the x-coordinate of D is 0 + b, or b, and the y-coordinate of D is 0 + c, or c.
A. C(c, c)
B. C(a, c)
C. C(a + b, c)
D. C(b, c)
Name the missing coordinates for the parallelogram.
Coordinate Proof
Place a rhombus on the coordinate plane. Label the midpoints of the sides M, N, P, and Q. Write a coordinate proof to prove that MNPQ is a rectangle.The first step is to position a rhombus on the coordinate plane so that the origin is the midpoint of the diagonals and the diagonals are on the axes, as shown. Label the vertices to make computations as simple as possible.Given: ABCD is a rhombus as labeled. M, N, P, Q are
midpoints.Prove: MNPQ is a rectangle.
Coordinate Proof
Proof:By the Midpoint Formula, the coordinates of M, N, P, and Q are as follows.
Find the slopes of
Coordinate Proof
Coordinate Proof
A segment with slope 0 is perpendicular to a segment with undefined slope. Therefore, consecutive sides of this quadrilateral are perpendicular. MNPQ is, by definition, a rectangle.
Place an isosceles trapezoid on the coordinate plane. Label the midpoints of the sides M, N, P, and Q. Write a coordinate proof to prove that MNPQ is a rhombus.
Given: ABCD is an isosceles trapezoid. M, N, P, and Q are midpoints.
Prove: MNPQ is a rhombus.
The coordinates of M are (–3a, b); the coordinates of N
are (0, 0); the coordinates of P are (3a, b); the
coordinates of Q are (0, 2b).
Since opposite sides have equal slopes, opposite sides
are parallel and MNPQ is a parallelogram. The slope of
The slope of is undefined. So, the diagonals
are perpendicular. Thus, MNPQ is a rhombus.
Proof:
A.
B.
C.
D.
Which expression would be the lengths of the four sides of MNPQ?
Properties of Quadrilaterals
Proof:
Since have the same slope, they are parallel.
Write a coordinate proof to prove that the supports of a platform lift are parallel.
Prove:
Given: A(5, 0), B(10, 5), C(5, 10), D(0, 5)
A. slopes = 2
B. slopes = –4
C. slopes = 4
D. slopes = –2
Prove:
Given: A(–3, 4), B(1, –4), C(–1, 4), D(3, –4)
Area of a RectangleA = bh Area = (Base)(Height)
h
b
Area of a ParallelogramA = bhBase and height must be
h
b
Area of a Triangle
2or
21 bhAbhA
Base and height must be
h
b
Area of a Trapezoid
hbbAbbhA2
)(or )(21 21
21
Bases and height must be
h
b1
b2
Area of a Kite
2))((or ))((
21 21
21ddAddA
d1
d2
Area of a Rhombus
2))((or ))((
21 21
21ddAddA
d1
d2
Homework Chapter 6.79th and 10th gradersPg 366: 7-14 & worksheet
distributed