2025 AMC 8 Problem 24

Question

In trapezoid

ABCD,

angles

B

and

C

measure

60^\circ

and

AB = DC.

The side lengths are all positive integers and the perimeter of

ABCD

is

30

units. How many non-congruent trapezoids satisfy all of these conditions?

0

1

2

3

4

Answer: E

Concepts:trapezoid equilateral triangle Diophantine Equation

Difficulty rating: 1840

Small hint:

Draw a line through

A

parallel to

CD

Big hint:

The perimeter becomes

3x+2y

for positive integers

x,y

Video solution:

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Written solution:

Draw a line through

A

parallel to line

CD,

and observe that some lengths are automatically equal to each other:

That is because

\angle AEB = \angle DCB = 60^\circ,

and so triangle

ABE

is equilateral. All lengths labeled

x

are always equal.

Also,

ADCE

is a parallelogram (not necessarily a rhombus), so the remaining lengths labeled

y

are always equal to each other, but not necessarily equal to

x.

The perimeter is

3x + 2y,

but it is also supposed to be

30.

The problem then amounts to counting how many positive integer solutions there are to the equation

3x + 2y = 30.

As we consider positive integers for

x,

observe that it is precisely the even integers which give an integer solution for

y,

because

30 - 3x

is the even number

2y.

And, once we get to

x = 10,

the

y

that satisfies the equation is

y = \frac{30 - 3 \times 10}{2} = 0,

which is not a positive integer.

So, the values that work for

x

are the positive even integers less than

10,

or

\{2, 4, 6, 8\}.

There are

4

options, which gives the answer of E.