Exercise 8 Page 71

a We can find the perimeter of each figure by counting the number of outermost sides they have. Consider the first figure.

We find its perimeter by counting the outermost sides.

Perimeter 1 + 1 + 1 + 1 = 4

The perimeter of the first figure is

4 .

By following the same procedure, the perimeters of figures

2

and

3

can be found.

Figure	Perimeter
$1$	$4$
$2$	$12$
$3$	$20$

b Let's try to find a pattern in the figures. We will distinguish between the types of sides with two different colors.

In the first figure there are only red sides, and no blue sides. Transitioning from Figure $1$ to Figure $2,$ the number of red edges remains $4,$ while we add $8$ blue edges.

Going from Figure

2

to Figure

3,

the number of red edges remains the same —

4,

while the number of blue edges increase by another

8 .

Using this observation, we can rewrite the perimeters of the first three figures.

Figure 1 4 + 0 \cdot 8 Figure 2 4 + 1 \cdot 8 Figure 3 4 + 2 \cdot 8

Examining the sequence we have created, we see that between two figures the perimeter increases by

8

units. This gives us a formula for the perimeter of the

n th

figure.

4 + (n - 1) \cdot 8 \Leftrightarrow 8 n - 12

To make sure that this is correct, lets substitute

n = 1,

2,

and

3

into the formula.

$n$	Substitute	Simplify
$1$	$8 (1) - 4$	$4$
$2$	$8 (2) - 4$	$12$
$3$	$8 (3) - 4$	$20$

The results are

4,

12,

and

20,

as expected.

c We have already found a formula for the perimeter of the

n th

figure. Let's use it to find the perimeter of Figure

10

by substituting

n = 10

into our formula.

8 n - 4

Substitute

$n = 10$

8 (10) - 4

Multiply

80 - 4

SubTerm

Subtract term

76

The perimeter of Figure

10

76 .

Subchapter links

Multiple Choice p.70

Short Response/Gridded Response p.71

Extended Response p.71