Transcript Geo Ch 6-7 – Coordinate Proof with Quadrilaterals

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 of
Quadrilaterals
• 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)
Quadrilaterals
Kites
Parallelograms
Rhombus Square
Rectangle
Trapezoids
Isosceles
Trapezoids
Tree Diagram
Quadrilateral
Parallelogram
Rhombus
Rectangle
Square
Kite
Trapezoid
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
1. Let A, B, C, and D be vertices of a rectangle with sides
a units long, and sides
b units
long.
2. 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.
3. 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.
Positioning a Square
4. D is on the y-axis so the x-coordinate is 0. Since the side
length is b, the y-coordinate is b.
5. The x-coordinate of C is also a. The y-coordinate is 0 + b
or b because the side
is b units long.
Sample answer:
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.
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.
Answer: D(b, c)
Name the missing coordinates
for the parallelogram.
A. C(c, c)
B. C(a, c)
C. C(a + b, c)
D. C(b, c)
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.
Proof:
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.
Which expression would be the lengths of the four
sides of MNPQ?
A.
B.
C.
D.
Properties of Quadrilaterals
Write a coordinate proof to
prove that the supports of a
platform lift are parallel.
Given: A(5, 0), B(10, 5),
C(5, 10), D(0, 5)
Prove:
Proof:
Since
parallel.
have the same slope, they are
Given: A(–3, 4), B(1, –4),
C(–1, 4), D(3, –4)
Prove:
A. slopes = 2
B. slopes = –4
C. slopes = 4
D. slopes = –2
Area of a Rectangle
A = bh
Area = (Base)(Height)
h
b
Area of a Parallelogram
A = bh
Base and height must be 
h
b
Area of a Triangle
1
bh
A  bh or A 
2
2
Base and height must be 
h
b
Area of a Trapezoid
1
(b1  b2 )
A  h(b1  b2 ) or A 
h
2
2
Bases and height must be 
b2
h
b1
Area of a Kite
1
(d1 )( d 2 )
A  (d1 )( d 2 ) or A 
2
2
d1
d2
Area of a Rhombus
1
(d1 )( d 2 )
A  (d1 )( d 2 ) or A 
2
2
d1
d2
Homework
Chapter 6.7
9th and 10th graders
Pg 366: 7-14 & worksheet
distributed