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.

Coordinate Proof with Quadrilaterals Lesson 6-7 Coordinate Proof with Quadrilaterals Chapter Menu

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 Lesson 7 Menu

Position and label quadrilaterals for use in coordinate proofs. Prove theorems using coordinate proofs. Lesson 7 MI/Vocab

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) Lesson 7 CA

Quadrilaterals Kites Parallelograms Trapezoids Rhombus Square Rectangle Square Isosceles Trapezoids Quadrilaterals

Tree Diagram Quadrilateral Parallelogram Kite Trapezoid Rhombus Rectangle Isosceles Trapezoid Square

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. Lesson 7 Ex1

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: Lesson 7 Ex1

Position and label a square with sides a units long on the coordinate plane. Which diagram would best achieve this? A. B. C. D. Lesson 7 CYP1

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) Lesson 7 Ex2

Name the missing coordinates for the parallelogram. A. C(c, c) B. C(a, c) C. C(a + b, c) D. C(b, c) Lesson 7 CYP2

Given: ABCD is a rhombus as labeled. M, N, P, Q are midpoints. 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. Lesson 7 Ex3

Coordinate Proof Proof: By the Midpoint Formula, the coordinates of M, N, P, and Q are as follows. Find the slopes of Lesson 7 Ex3

Coordinate Proof Lesson 7 Ex3

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. Lesson 7 Ex3

Given: ABCD is an isosceles trapezoid. M, N, P, and Q are midpoints. 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. Lesson 7 CYP3

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: Lesson 7 CYP3

Which expression would be the lengths of the four sides of MNPQ? A. B. C. D. Lesson 7 CYP3

Properties of Quadrilaterals 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) Proof: Since have the same slope, they are parallel. Lesson 7 Ex4

Given: A(–3, 4), B(1, –4), C(–1, 4), D(3, –4) Prove: Given: A(–3, 4), B(1, –4), C(–1, 4), D(3, –4) A. slopes = 2 B. slopes = –4 C. slopes = 4 D. slopes = –2 Lesson 7 CYP4

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 Base and height must be  h b

Area of a Trapezoid Bases and height must be  h b1 b2

Area of a Kite d1 d2

Area of a Rhombus d1 d2

Homework Chapter 6.7 9th and 10th graders Pg 366: 7-14 & worksheet distributed