Equations: Perimeter/Angles Textbook Exercise - Corbettmaths - Free Printable

To solve the problem, we need to use the formula for the perimeter of a rectangle. The perimeter \( P \) of a rectangle is given by:

\[
P = 2 \times (\text{length} + \text{width})
\]

We will solve each part step by step.

(a) Perimeter = 24 cm

Given:
- Length = \( x + 6 \)
- Width = \( x \)

The perimeter formula is:
\[
24 = 2 \times ((x + 6) + x)
\]

Simplify inside the parentheses:
\[
24 = 2 \times (2x + 6)
\]

Distribute the 2:
\[
24 = 4x + 12
\]

Subtract 12 from both sides:
\[
12 = 4x
\]

Divide by 4:
\[
x = 3
\]

Now, find the length and width:
- Length = \( x + 6 = 3 + 6 = 9 \)
- Width = \( x = 3 \)

(b) Perimeter = 56 cm

Given:
- Length = \( x + 12 \)
- Width = \( x \)

The perimeter formula is:
\[
56 = 2 \times ((x + 12) + x)
\]

Simplify inside the parentheses:
\[
56 = 2 \times (2x + 12)
\]

Distribute the 2:
\[
56 = 4x + 24
\]

Subtract 24 from both sides:
\[
32 = 4x
\]

Divide by 4:
\[
x = 8
\]

Now, find the length and width:
- Length = \( x + 12 = 8 + 12 = 20 \)
- Width = \( x = 8 \)

(c) Perimeter = 88 cm

Given:
- Length = \( 3x \)
- Width = \( x \)

The perimeter formula is:
\[
88 = 2 \times (3x + x)
\]

Simplify inside the parentheses:
\[
88 = 2 \times 4x
\]

Distribute the 2:
\[
88 = 8x
\]

Divide by 8:
\[
x = 11
\]

Now, find the length and width:
- Length = \( 3x = 3 \times 11 = 33 \)
- Width = \( x = 11 \)

(d) Perimeter = 38 cm

Given:
- Length = \( 2x + 3 \)
- Width = \( x + 1 \)

The perimeter formula is:
\[
38 = 2 \times ((2x + 3) + (x + 1))
\]

Simplify inside the parentheses:
\[
38 = 2 \times (3x + 4)
\]

Distribute the 2:
\[
38 = 6x + 8
\]

Subtract 8 from both sides:
\[
30 = 6x
\]

Divide by 6:
\[
x = 5
\]

Now, find the length and width:
- Length = \( 2x + 3 = 2 \times 5 + 3 = 10 + 3 = 13 \)
- Width = \( x + 1 = 5 + 1 = 6 \)

(e) Perimeter = 158 cm

Given:
- Length = \( 5x + 3 \)
- Width = \( 2x + 6 \)

The perimeter formula is:
\[
158 = 2 \times ((5x + 3) + (2x + 6))
\]

Simplify inside the parentheses:
\[
158 = 2 \times (7x + 9)
\]

Distribute the 2:
\[
158 = 14x + 18
\]

Subtract 18 from both sides:
\[
140 = 14x
\]

Divide by 14:
\[
x = 10
\]

Now, find the length and width:
- Length = \( 5x + 3 = 5 \times 10 + 3 = 50 + 3 = 53 \)
- Width = \( 2x + 6 = 2 \times 10 + 6 = 20 + 6 = 26 \)

(f) Perimeter = 60 cm

Given:
- Length = \( x + 9 \)
- Width = \( x - 3 \)

The perimeter formula is:
\[
60 = 2 \times ((x + 9) + (x - 3))
\]

Simplify inside the parentheses:
\[
60 = 2 \times (2x + 6)
\]

Distribute the 2:
\[
60 = 4x + 12
\]

Subtract 12 from both sides:
\[
48 = 4x
\]

Divide by 4:
\[
x = 12
\]

Now, find the length and width:
- Length = \( x + 9 = 12 + 9 = 21 \)
- Width = \( x - 3 = 12 - 3 = 9 \)

(g) Perimeter = 2 m

Given:
- Length = \( 2x - 9 \)
- Width = \( x + 1 \)

The perimeter formula is:
\[
2 = 2 \times ((2x - 9) + (x + 1))
\]

Simplify inside the parentheses:
\[
2 = 2 \times (3x - 8)
\]

Distribute the 2:
\[
2 = 6x - 16
\]

Add 16 to both sides:
\[
18 = 6x
\]

Divide by 6:
\[
x = 3
\]

Now, find the length and width:
- Length = \( 2x - 9 = 2 \times 3 - 9 = 6 - 9 = -3 \) (This is not possible for a length, so there might be an error in the problem setup or interpretation.)

(h) Perimeter = 82 cm

Given:
- Length = \( 6x + 1 \)
- Width = \( 10 - x \)

The perimeter formula is:
\[
82 = 2 \times ((6x + 1) + (10 - x))
\]

Simplify inside the parentheses:
\[
82 = 2 \times (5x + 11)
\]

Distribute the 2:
\[
82 = 10x + 22
\]

Subtract 22 from both sides:
\[
60 = 10x
\]

Divide by 10:
\[
x = 6
\]

Now, find the length and width:
- Length = \( 6x + 1 = 6 \times 6 + 1 = 36 + 1 = 37 \)
- Width = \( 10 - x = 10 - 6 = 4 \)

(i) Perimeter = 40 cm

Given:
- Length = \( 3x + 4 \)
- Width = \( 2x - 1 \)

The perimeter formula is:
\[
40 = 2 \times ((3x + 4) + (2x - 1))
\]

Simplify inside the parentheses:
\[
40 = 2 \times (5x + 3)
\]

Distribute the 2:
\[
40 = 10x + 6
\]

Subtract 6 from both sides:
\[
34 = 10x
\]

Divide by 10:
\[
x = 3.4
\]

Now, find the length and width:
- Length = \( 3x + 4 = 3 \times 3.4 + 4 = 10.2 + 4 = 14.2 \)
- Width = \( 2x - 1 = 2 \times 3.4 - 1 = 6.8 - 1 = 5.8 \)

Final Answers

\[
\boxed{
\begin{array}{ll}
(a) & x = 3, \text{Length} = 9, \text{Width} = 3 \\
(b) & x = 8, \text{Length} = 20, \text{Width} = 8 \\
(c) & x = 11, \text{Length} = 33, \text{Width} = 11 \\
(d) & x = 5, \text{Length} = 13, \text{Width} = 6 \\
(e) & x = 10, \text{Length} = 53, \text{Width} = 26 \\
(f) & x = 12, \text{Length} = 21, \text{Width} = 9 \\
(g) & \text{Not possible with real dimensions} \\
(h) & x = 6, \text{Length} = 37, \text{Width} = 4 \\
(i) & x = 3.4, \text{Length} = 14.2, \text{Width} = 5.8 \\
\end{array}
}
\]