I want to fence in a rectangular garden next to a wall of my house. I have a total of 20 meters of chicken wire to use to construct the fence. I only need to construct three sides of the rectangle, since the wall of my house will serve as the fourth side.

What is the maximum possible area for the garden?

Back wall of the house

w w

20 – 2w

Area = w(20 – 2w) = 20w – 2w2

w A

1

2

3

4

5

6

7

8

9

18

32

48

48

42

42

32

50

18

M

LK

Q

R

P

B

A

C

3

1

3

1

3

1

1. In the last homework assignment, you constructed and saved the diagram shown, in which AP = AB, CR = AC,

a. Display the area of ABC and KLM (the shaded triangle).

b. Make a conjecture about how many times larger the area ABC is than the area of KLM.

BQ = BC

The area of ABC is 7 times the area of KLM.

From the Homework

2. a. Using Geometer’s Sketchpad, construct a regular octagon. b. Construct the midpoint of the longest diagonal and call it point P (as shown). c. Using point P as center, construct a circle that passes through each vertex of the octagon. We say that the octagon is inscribed in the circle. d. Construct a segment ( ) from the circle’s center perpendicular to a side of the

octagon. Note, is called an apothem of the regular octagon.

e. Display the area of the octagon.

f. By displaying the appropriate measurements, verify that the area of the octagon is equal to one-half the product of the apothem and the perimeter. [ i.e. Area = (apothem)(perimeter) or A = ap ]

PRPR

2

12

1

R

P

3. Using Geometer’s Sketchpad, construct a rhombus and its two diagonals. a. Display the lengths of both diagonals

b. Display the area of the rhombus.

c. Conjecture a formula for the area of a rhombus in terms of the product of the lengths of the diagonals.

d. Prove your conjecture.

A = ½ (d1)(d2)

AB

CD

4. Experiment on Geometer’s Sketchpad to find a way to obtain the area of

a. a circle b. a sector of a circle. c. a segment of a circle.

sector segment

8.5

94

B

AC

1315

14F

E

G

H

7

11

4M L

J K

N

8. In the parallelogram from question 7 what is the length of the altitude from H to ?

FG

5. ABC 6. Parallelogram EFGH

Find the area of each figure indicated below (questions 6-8):

7. Isosceles Trapezoid JKLM

55.25 195

105 or 13.92857

14

1313

?

9. In the diagram, ABCD is a rectangle. Point P is chosen randomly on side . Conjecture a relationship between the area of the rectangle and the area of APB (shaded). Prove that your conjecture is true. (Note: use of Sketchpad on this question is optional

DC

C

A B

D P

))()(( cSbSaSS

15

14

13

10. Use the Geometer’s Sketchpad construction of the 13-14-15 triangle (sent to you earlier) to complete this question.  a. Use the area option on Geometer’s Sketchpad to obtain the area of triangle ABC.

b. Construct an altitude from C to AB and use the formulas A = ½bh to verify the area you found in part (a).

c. Heron’s formula for the area of a triangle states that the area of triangle with side lengths a, b, and c can be obtained using the formula

Area = where S is half the perimeter of the triangle (S is called the semi-perimeter, and Heron’s formula is often referred to as the semi-perimeter formula). Use Heron’s formula to verify the area you obtained in parts (a) and (b).

12

 11.Use Geometer’s Sketchpad to construct a parallelogram whose area does not change when its vertices are dragged. (Hint: look back at question 9.)

Using Geometer’s Sketchpad, construct a circle. Construct a diameter of the circle. Display the length of the diameter and the circumference of the circle accurate to three decimal places.

Circumference AB = 26.567 cm

BC = 8.456 cm

C

A

B