Compare Trapezoids: Are Two Trapezoids with Sides x and x+16 Congruent?

Congruent Trapezoids with Equal Perimeters

Given the trapezoids in the drawing.

Are they the same trapezoid?

CCCxx+1610677

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Are the trapezoids equal?
00:03 Let's mark trapezoids 1,2
00:06 The perimeter of a trapezoid equals the sum of its sides
00:10 Let's calculate the perimeter of trapezoid 1
00:21 This is the perimeter of trapezoid 1
00:24 Let's use the same method to calculate the perimeter of trapezoid 2
00:39 This is the size of trapezoid 2's perimeter and it equals trapezoid 1
00:46 The trapezoids have equal perimeter
00:50 But they are different because trapezoid 2 is right-angled
00:55 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

Given the trapezoids in the drawing.

Are they the same trapezoid?

CCCxx+1610677

2

Step-by-step solution

We calculate the perimeter of the left trapezoid:

P=6+10+7+52x+5 P=6+10+7+\frac{5}{2}x+5

P=28+52x P=28+\frac{5}{2}x

We calculate the perimeter of the right trapezoid:

P=7+x+x+16+x2+5 P=7+x+x+16+\frac{x}{2}+5

P=212x+28 P=2\frac{1}{2}x+28

P=52x+28 P=\frac{5}{2}x+28

The perimeters of the two trapezoids are equal to each other.

3

Final Answer

No, but their perimeter is identical.

Key Points to Remember

Essential concepts to master this topic
  • Congruence: Same shape AND size means all sides and angles match
  • Technique: Calculate each perimeter: P=6+10+7+5x2+5=28+5x2 P = 6+10+7+\frac{5x}{2}+5 = 28+\frac{5x}{2}
  • Check: Equal perimeters don't guarantee congruence - verify all corresponding sides ✓

Common Mistakes

Avoid these frequent errors
  • Assuming equal perimeters means congruent shapes
    Don't think that identical perimeters = congruent trapezoids! Two shapes can have the same perimeter but completely different side lengths and angles. Always check if ALL corresponding sides and angles are equal for true congruence.

Practice Quiz

Test your knowledge with interactive questions

Given the trapezoid:

999121212555AAABBBCCCDDDEEE

What is the area?

FAQ

Everything you need to know about this question

If the perimeters are the same, why aren't the trapezoids congruent?

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Congruent shapes must have identical corresponding sides and angles, not just equal perimeters. Think of it like this: a rectangle with sides 3×7 and another with sides 1×9 both have perimeter 20, but they're completely different shapes!

How do I calculate the perimeter when there are variables?

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Add up all four sides just like with numbers! For the left trapezoid: 6+10+7+5x2+5=28+5x2 6 + 10 + 7 + \frac{5x}{2} + 5 = 28 + \frac{5x}{2} . Combine like terms and keep variables in the final expression.

What does it mean that both trapezoids have the same perimeter?

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It means if you walked around the edge of either trapezoid, you'd travel the same total distance. Both perimeters equal 28+5x2 28 + \frac{5x}{2} , but the shapes are arranged differently.

Could these trapezoids ever be congruent for some value of x?

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No! Looking at the diagrams, the trapezoids have different orientations and side arrangements. Even though their perimeters are equal, their corresponding sides don't match up for any value of x.

How can I tell if two shapes are congruent?

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  • All corresponding sides must be equal
  • All corresponding angles must be equal
  • Having the same area or perimeter alone is not enough

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