Find Rectangle's Side Length: Given Area 21 cm² and Side Sum 14 cm

Rectangle Dimensions with Area Constraints

What is the length of the second side in a rectangle, given that the sum of the 2 opposite sides is 14 cm and the area of the rectangle is 21 cm²?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Determine the length of the second side in the rectangle
00:03 Draw the rectangle
00:07 In a rectangle, opposite sides are equal
00:12 The sum of opposite sides according to the given data
00:17 Isolate X
00:21 This is the length of side X in the rectangle
00:28 In a rectangle opposite sides are equal, let's mark it as Y
00:32 The area of the rectangle according to the given data
00:35 Apply the formula for calculating the rectangle area (side x side)
00:38 Substitute in the relevant values according to the data and solve for Y
00:41 Isolate Y
00:46 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

What is the length of the second side in a rectangle, given that the sum of the 2 opposite sides is 14 cm and the area of the rectangle is 21 cm²?

2

Step-by-step solution

As we know that the two opposite sides of the rectangle add up to 14 and that, in a rectangle, each pair of opposite sides are equal, we can calculate the size of each one as follows:

14:2=7 14:2=7

We will call the other two opposite sides y y and calculate the unknown variable as follows:

7×y=21 7\times y=21

We will then divide both sides by 7 to get:

y=3 y=3

3

Final Answer

3

Key Points to Remember

Essential concepts to master this topic
  • Rectangle Properties: Opposite sides are equal, so length pairs are identical
  • System Setup: Use l+w=7 l + w = 7 and l×w=21 l \times w = 21
  • Verification: Check that 7×3=21 7 \times 3 = 21 and 7+3=1014 7 + 3 = 10 \neq 14

Common Mistakes

Avoid these frequent errors
  • Confusing perimeter with side sum
    Don't assume the sum of 2 opposite sides equals half the perimeter = wrong setup! The problem states sum of 2 opposite sides is 14, not that perimeter is 28. Always read carefully: sum of opposite sides means one length plus one width.

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle ABCD below.

Side AB is 6 cm long and side BC is 4 cm long.

What is the area of the rectangle?
666444AAABBBCCCDDD

FAQ

Everything you need to know about this question

Why do we divide 14 by 2 to get 7?

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Since opposite sides are equal in a rectangle, if two opposite sides sum to 14, each of those sides must be 14÷2=7 14 \div 2 = 7 cm long.

How do I know which side is 7 and which is 3?

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It doesn't matter! The problem asks for the length of the second side. Since we found one pair of sides is 7 cm, the other pair must be 3 cm.

Can I solve this using a quadratic equation?

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Yes! Let one side be x x . Then: x(7x)=21 x(7-x) = 21 , which gives x27x+21=0 x^2 - 7x + 21 = 0 . But the direct method shown is simpler!

What if I get a decimal answer?

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Rectangle problems can have decimal dimensions! Always check your work by verifying both the area and the given side relationship.

Why can't the answer be 7?

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If the second side were 7 cm, we'd have a square with area 7×7=49 7 \times 7 = 49 cm², not 21 cm² as required.

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