Rectangle Perimeter with 4×8 Dimensions and Embedded Equilateral Triangle

Rectangle Perimeter with Equilateral Triangle Properties

Look at the following rectangle:

AAABBBCCCDDDEEE84

ΔEDC is equilateral.

Calculate the perimeter of the rectangle.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find the perimeter of the rectangle
00:03 Equilateral triangle according to the given data
00:06 Opposite sides in a rectangle are equal
00:20 The perimeter of the rectangle equals the sum of its sides
00:32 Let's solve every 2 factors separately and then sum them
00:39 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Look at the following rectangle:

AAABBBCCCDDDEEE84

ΔEDC is equilateral.

Calculate the perimeter of the rectangle.

2

Step-by-step solution

A rectangle has two pairs of equal opposite sides.

That is:

BC=AD=4

AB=DC

In an equilateral triangle, all sides are equal, therefore:
EC=CD=DE

We know that EC=8, so:

EC=CD=DE=8

We know that:

AB=DC

Therefore:

AB=DC=8

Remember that the perimeter of a rectangle is equal to the sum of all its sides, therefore:

AB+BC+DC+AD

We substitute in all its known sides:

8+4+8+4=

24

3

Final Answer

24

Key Points to Remember

Essential concepts to master this topic
  • Rectangle Property: Opposite sides are equal in length
  • Equilateral Triangle: All three sides equal, so EC = CD = DE = 8
  • Check: Perimeter = AB + BC + CD + AD = 8 + 4 + 8 + 4 = 24 ✓

Common Mistakes

Avoid these frequent errors
  • Using triangle side labels instead of rectangle properties
    Don't assume DC = 4 just because BC = 4 = wrong rectangle property! This ignores that triangle EDC is equilateral with all sides = 8. Always use the equilateral triangle property: EC = CD = DE = 8, then apply rectangle property AB = DC.

Practice Quiz

Test your knowledge with interactive questions

Calculate the perimeter of the rectangle below.

181818222

FAQ

Everything you need to know about this question

How do I know which sides of the rectangle are equal?

+

In any rectangle, opposite sides are equal. So AB = DC and BC = AD. Look at the diagram carefully to identify which sides are opposite to each other.

Why does the equilateral triangle matter for finding the rectangle's perimeter?

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The equilateral triangle EDC tells us that all its sides are equal. Since EC = 8, then CD = 8 too. This gives us the length of one side of the rectangle!

What if I can't see all the measurements clearly?

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Use the given information strategically! You know BC = 4 from the diagram, and EC = 8. Since triangle EDC is equilateral, CD must also equal 8.

How do I calculate the perimeter once I know all sides?

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Add up all four sides of the rectangle: Perimeter = AB + BC + CD + AD. Since AB = CD = 8 and BC = AD = 4, you get: 8 + 4 + 8 + 4 = 24.

Could the answer be different if the triangle wasn't equilateral?

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Yes! If triangle EDC wasn't equilateral, CD might not equal 8. The equilateral property is crucial for determining that CD = EC = 8.

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