Isosceles Trapezoid Problem: Finding C2D2 Length in Equal Shapes

Q: Why do equal trapezoids have equal perimeters?

Equal shapes means they are congruent - identical in every way! This means all corresponding sides match, so their perimeters must be the same.

Q: How do I identify which sides to add for perimeter?

For any trapezoid, add all four sides: top base + bottom base + left side + right side. In isosceles trapezoids, the two slanted sides are always equal.

Q: What if I get a negative answer?

Check your algebra! In this problem, $ C_2D_2 = 2X - 4 $ could be negative if X is small. That's mathematically correct, but make sure it makes sense in context.

Q: Can I solve this without setting up an equation?

Not easily! Setting up the equation perimeter₁ = perimeter₂ is the most reliable method. It clearly shows the relationship between all the given measurements.

Q: How do I know which trapezoid is which?

Look at the labels! The first trapezoid has vertices $ A_1B_1C_1D_1 $ and the second has $ A_2B_2C_2D_2 $. We're finding the length of side $ C_2D_2 $ in the second trapezoid.

Equal Trapezoid Perimeters with Unknown Sides

The two trapezoids below are isosceles.

What is the size of C2D2 if the trapezoids are equal?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find side C2D2

00:03 Let's start by calculating the perimeter of trapezoid 1

00:07 The perimeter of the trapezoid equals the sum of its sides

00:10 Let's substitute appropriate values according to the given data and solve for the perimeter

00:15 This is the perimeter size of trapezoid 1

00:18 Now we'll use this size to find side C2D2

00:22 Let's substitute appropriate values in the perimeter formula and solve for the side

00:38 Let's substitute the perimeter size in the equation to find the side

00:46 Let's isolate side C2D2

01:05 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The two trapezoids below are isosceles.

What is the size of C2D2 if the trapezoids are equal?

Step-by-step solution

To solve this problem, let's calculate the perimeters of both trapezoids:

For the first trapezoid $A_1B_1C_1D_1$ :

Top base: $A_1B_1 = 2X$
Bottom base: $C_1D_1 = 3X$
Two equal side extensions, each of length 4.

The perimeter $P_1$ of the first trapezoid is:
$P_1 = 2X + 3X + 4 + 4 = 5X + 8$ .

For the second trapezoid $A_2B_2C_2D_2$ :

Top base: $A_2B_2 = 3X$
Bottom base: $C_2D_2 = ?$
Two equal side extensions, each of length 6.

The perimeter $P_2$ of the second trapezoid is:
$P_2 = 3X + C_2D_2 + 6 + 6 = 3X + C_2D_2 + 12$ .

Since the trapezoids are equal, their perimeters are the same:
$5X + 8 = 3X + C_2D_2 + 12$ .

Solving for $C_2D_2$ :

$C_2D_2 = 5X + 8 - 3X - 12$

$C_2D_2 = 2X - 4$

Thus, the size of $C_2D_2$ if the trapezoids are equal is $2X - 4$ .

Final Answer

2X-4

Key Points to Remember

Essential concepts to master this topic

Equal Shapes: Same trapezoids have identical perimeters in all cases
Technique: Set up equation: $5X + 8 = 3X + C_2D_2 + 12$
Check: Verify both perimeters equal same value when substituting answer ✓

Common Mistakes

Avoid these frequent errors

Adding areas instead of perimeters
Don't calculate area by multiplying bases and heights = completely wrong approach! Equal shapes means equal perimeters, not equal areas. Always add all four sides to find the perimeter.

Practice Quiz

Test your knowledge with interactive questions

Calculate the perimeter of the trapezoid according to the following data:

FAQ

Everything you need to know about this question

Why do equal trapezoids have equal perimeters?

Equal shapes means they are congruent - identical in every way! This means all corresponding sides match, so their perimeters must be the same.

How do I identify which sides to add for perimeter?

For any trapezoid, add all four sides: top base + bottom base + left side + right side. In isosceles trapezoids, the two slanted sides are always equal.

What if I get a negative answer?

Check your algebra! In this problem, $C_2D_2 = 2X - 4$ could be negative if X is small. That's mathematically correct, but make sure it makes sense in context.

Can I solve this without setting up an equation?

Not easily! Setting up the equation perimeter₁ = perimeter₂ is the most reliable method. It clearly shows the relationship between all the given measurements.

How do I know which trapezoid is which?

Look at the labels! The first trapezoid has vertices $A_1B_1C_1D_1$ and the second has $A_2B_2C_2D_2$ . We're finding the length of side $C_2D_2$ in the second trapezoid.

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