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