Question

question 11
a regular octagon with sides of length 7 and an apothem of length 10.49 has an area of ______ square units.
a. 73.43
b. 293.72
c. 92.39
d. 440.16
e. 79.25
f. 587.44

question 12
what is the area of the polygon below?
a. 156 square units
b. 111 square units
c. 147 square units
d. 120 square units

question 13
what is the equation of the line containing the points (3, 1), (9, 3), and (27, 9)?
a. $y = 3x$
b. $y = \frac{1}{3}x$
c. $y = x^3$

question 14
which of the following points are on the line given by the equation $y = \frac{1}{2}x$?
check all that apply.
a. (-2, -1)
b. (-2, 1)
c. (3, 15)
d. (2, 1)
e. (3, 6)
f. (4, 2)

Explanation:

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Question 11

Step1: Recall regular polygon area formula

Area = $\frac{1}{2} \times \text{perimeter} \times \text{apothem}$

Step2: Calculate perimeter of octagon

Perimeter = $8 \times 7 = 56$

Step3: Substitute values into formula

Area = $\frac{1}{2} \times 56 \times 10.49 = 28 \times 10.49 = 293.72$

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Question 12

Step1: Split polygon into rectangles

We can split the shape into a large rectangle and a smaller rectangle (or subtract a small rectangle from a full rectangle). Using the subtraction method: full rectangle dimensions are $13 \times 12$, missing rectangle dimensions are $(13-10) \times 3 = 3 \times 3$.

Step2: Calculate full rectangle area

Area of full rectangle = $13 \times 12 = 156$

Step3: Calculate missing rectangle area

Area of missing rectangle = $3 \times 3 = 9$

Step4: Subtract to get polygon area

Area = $156 - 9 = 147$

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Question 13

Step1: Calculate slope of the line

Slope $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3-1}{9-3} = \frac{2}{6} = \frac{1}{3}$

Step2: Verify with another point

Check $(27,9)$: $m = \frac{9-3}{27-9} = \frac{6}{18} = \frac{1}{3}$

Step3: Find line equation (y-intercept=0)

Using point $(3,1)$: $1 = \frac{1}{3} \times 3 + b \implies b=0$, so $y = \frac{1}{3}x$

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Question 14

Step1: Test point A $(-2,-1)$

Substitute $x=-2$: $y = \frac{1}{2} \times (-2) = -1$, matches.

Step2: Test point B $(-2,1)$

Substitute $x=-2$: $y = \frac{1}{2} \times (-2) = -1
eq 1$, no match.

Step3: Test point C $(3,15)$

Substitute $x=3$: $y = \frac{1}{2} \times 3 = 1.5
eq 15$, no match.

Step4: Test point D $(2,1)$

Substitute $x=2$: $y = \frac{1}{2} \times 2 = 1$, matches.

Step5: Test point E $(3,6)$

Substitute $x=3$: $y = \frac{1}{2} \times 3 = 1.5
eq 6$, no match.

Step6: Test point F $(4,2)$

Substitute $x=4$: $y = \frac{1}{2} \times 4 = 2$, matches.

Answer:

1. Question 11: B. 293.72
2. Question 12: C. 147 square units
3. Question 13: B. $y = \frac{1}{3}x$
4. Question 14: A. $(-2,-1)$, D. $(2,1)$, F. $(4,2)$