Lesson 8-7

Coordinate Proof with Quadrilaterals

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ABCD is an isosceles trapezoid with median EF.

1. Find mD if mA= 110°.

2. Find x if AD = 3x² + 5 and BC = x² + 27.

3. Find y if AC = 9(2y – 4) and BD = 10y + 12.

4. Find EF if AB = 10 and CD = 32.

5. Find AB if AB = r + 18, CD = 6r + 9 and EF = 4r + 10.

6. Which statement is always true about trapezoid LMNO with bases of LM and NO?Standardized Test Practice:

A

C

B

D

LO // MN LO MN

A B

D C

FE

LM NOLM // NO

70°

± 4

6

21

25

C

Objectives

• Position and label quadrilaterals for use in coordinate proofs

• Prove theorems using coordinate proofs

Vocabulary

• Kite – quadrilateral with exactly two distinct pairs of adjacent congruent sides.

Polygon Hierarchy

Polygons

Squares

RhombiRectangles

Parallelograms Kites Trapezoids

IsoscelesTrapezoids

Quadrilaterals

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 and the y-coordinate of D is 0 + c, or c0 + b, or b.

Answer: D (b, c)

A(0, 0) B(a, 0)

D(?, ?) C(a-b, c)y

x

Name the missing coordinates for the rhombus.

Answer:

Quadrilateral Characteristics Summary

Convex Quadrilaterals

Squares

RhombiRectangles

Parallelograms Trapezoids

IsoscelesTrapezoids

Opposite sides parallel and congruentOpposite angles congruentConsecutive angles supplementaryDiagonals bisect each other

Bases ParallelLegs are not ParallelLeg angles are supplementary Median is parallel to basesMedian = ½ (base + base)

Angles all 90°Diagonals congruent

Diagonals divide into 4 congruent triangles

All sides congruentDiagonals perpendicularDiagonals bisect opposite angles

Legs are congruent Base angle pairs congruent Diagonals are congruent

4 sided polygon4 interior angles sum to 3604 exterior angles sum to 360

Do you know your characteristics?

• Extra Credit Assignment

• Review Problems

In the isosceles trapezoid to the rightEF is a median, solve for x, y and z

2x + 8

6x - 6

21 2y - 4

y + 4 6z

12zA B

C D

E F

x = 5y = 8z = 10

In the rhombus to the left, JK = 6x, KM = 2y, LNM = 10y,

JLN = 4z + 10, and JKN = 7z – 5, solve for x, y and z

J K

L M

N

x = 3y = 9z = 5

In the rectangle to the left, WA = 6x, AH = 24, AHB = 33°, WAP = y, and BAP = z – 5,

solve for x, y and z

W P

BH

A x = 4y = 114°z = 71°

In the square to the right, RV = 5x, SV = 3y, VST = 9y,

and RS = z solve for x, y and z

R S

TU

V x = 3y = 5z = 15√2

Summary & Homework

• Summary:– Position a quadrilateral so that a vertex is at the

origin and a least one side lies along an axis.

• Homework: – pg 450-451; 9, 11-14, 28, 29, 31-33