Here are the step-by-step solutions for each rectangle.
The Rule: The perimeter of a rectangle is found by adding all four sides together.
Formula: $2 \times (\text{Length} + \text{Width}) = \text{Perimeter}$
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(a) Perimeter = 24cm
Sides are $x$ and $x + 6$.
1. Equation: $2(x + x + 6) = 24$
2. Simplify inside brackets: $2(2x + 6) = 24$
3. Divide by 2: $2x + 6 = 12$
4. Subtract 6: $2x = 6$
5. Divide by 2:
$x = 3$
* Width ($x$):
3 cm
* Length ($x + 6$): $3 + 6 =$
9 cm
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(b) Perimeter = 56cm
Sides are $x$ and $x + 12$.
1. Equation: $2(x + x + 12) = 56$
2. Simplify: $2(2x + 12) = 56$
3. Divide by 2: $2x + 12 = 28$
4. Subtract 12: $2x = 16$
5. Divide by 2:
$x = 8$
* Width ($x$):
8 cm
* Length ($x + 12$): $8 + 12 =$
20 cm
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(c) Perimeter = 88cm
Sides are $x$ and $3x$.
1. Equation: $2(x + 3x) = 88$
2. Simplify: $2(4x) = 88 \rightarrow 8x = 88$
3. Divide by 8:
$x = 11$
* Width ($x$):
11 cm
* Length ($3x$): $3 \times 11 =$
33 cm
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(d) Perimeter = 38cm
Sides are $2x + 3$ and $x + 1$.
1. Equation: $2((2x + 3) + (x + 1)) = 38$
2. Combine like terms inside: $2(3x + 4) = 38$
3. Divide by 2: $3x + 4 = 19$
4. Subtract 4: $3x = 15$
5. Divide by 3:
$x = 5$
* Side 1 ($2x + 3$): $(2 \times 5) + 3 =$
13 cm
* Side 2 ($x + 1$): $5 + 1 =$
6 cm
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(e) Perimeter = 158cm
Sides are $5x + 3$ and $2x + 6$.
1. Equation: $2((5x + 3) + (2x + 6)) = 158$
2. Combine like terms inside: $2(7x + 9) = 158$
3. Divide by 2: $7x + 9 = 79$
4. Subtract 9: $7x = 70$
5. Divide by 7:
$x = 10$
* Side 1 ($5x + 3$): $(5 \times 10) + 3 =$
53 cm
* Side 2 ($2x + 6$): $(2 \times 10) + 6 =$
26 cm
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(f) Perimeter = 60cm
Sides are $x + 9$ and $x - 3$.
1. Equation: $2((x + 9) + (x - 3)) = 60$
2. Combine like terms inside: $2(2x + 6) = 60$
3. Divide by 2: $2x + 6 = 30$
4. Subtract 6: $2x = 24$
5. Divide by 2:
$x = 12$
* Side 1 ($x + 9$): $12 + 9 =$
21 cm
* Side 2 ($x - 3$): $12 - 3 =$
9 cm
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(g) Perimeter = 2m
*Note: Since the algebra uses simple numbers, we will solve for $x$ in meters.*
Sides are $2x - 9$ and $x + 1$.
1. Equation: $2((2x - 9) + (x + 1)) = 2$
2. Combine like terms inside: $2(3x - 8) = 2$
3. Divide by 2: $3x - 8 = 1$
4. Add 8: $3x = 9$
5. Divide by 3:
$x = 3$
* Side 1 ($2x - 9$): $(2 \times 3) - 9 = 6 - 9 = -3$.
*(Note: In geometry, lengths cannot be negative. This suggests there might be a typo in the question's numbers, but mathematically $x=3$. If we assume the unit was meant to be different or the constant smaller, the method remains the same. Based strictly on the algebra provided:)*
* Math Answer for $x$:
3
* Calculated Sides:
-3 m and
4 m *(This indicates an error in the problem statement itself, as a side length cannot be negative).*
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(h) Perimeter = 82cm
Sides are $6x + 1$ and $10 - x$.
1. Equation: $2((6x + 1) + (10 - x)) = 82$
2. Combine like terms inside: $2(5x + 11) = 82$
3. Divide by 2: $5x + 11 = 41$
4. Subtract 11: $5x = 30$
5. Divide by 5:
$x = 6$
* Side 1 ($6x + 1$): $(6 \times 6) + 1 =$
37 cm
* Side 2 ($10 - x$): $10 - 6 =$
4 cm
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(i) Perimeter = 40cm
Sides are $3x + 4$ and $2x - 1$.
1. Equation: $2((3x + 4) + (2x - 1)) = 40$
2. Combine like terms inside: $2(5x + 3) = 40$
3. Divide by 2: $5x + 3 = 20$
4. Subtract 3: $5x = 17$
5. Divide by 5:
$x = 3.4$
* Side 1 ($3x + 4$): $(3 \times 3.4) + 4 = 10.2 + 4 =$
14.2 cm
* Side 2 ($2x - 1$): $(2 \times 3.4) - 1 = 6.8 - 1 =$
5.8 cm
Final Answer:
(a) x = 3; Length = 9cm, Width = 3cm
(b) x = 8; Length = 20cm, Width = 8cm
(c) x = 11; Length = 33cm, Width = 11cm
(d) x = 5; Length = 13cm, Width = 6cm
(e) x = 10; Length = 53cm, Width = 26cm
(f) x = 12; Length = 21cm, Width = 9cm
(g) x = 3; (Problem contains an error resulting in a negative side length of -3m and 4m)
(h) x = 6; Length = 37cm, Width = 4cm
(i) x = 3.4; Length = 14.2cm, Width = 5.8cm
Parent Tip: Review the logic above to help your child master the concept of area equations worksheet.
