2025 AMC 8 Problem 24

Right Triangle Decomposition Method

By dropping altitudes from the top vertices, we can split the trapezoid into a rectangle and two $30^{\circ }-60^{\circ }-90^{\circ }$ right triangles. This allows us to express all four side lengths using just two variables.

Step 1: Assign variables to the unknown sides

Let the length of the top base be represented by one variable, and let the lengths of the equal slanted sides be represented by another. Since the problem explicitly states that all side lengths are positive integers, both of these variables must be integers greater than or equal to 1.

Step 2: Draw altitudes to the base

Draw perpendicular lines from the top vertices down to the bottom base. Let the points where they intersect the bottom base be named $E$ and $F$.

Step 3: Analyze the resulting right triangles

This geometric construction creates two right triangles on the outer edges of the trapezoid. Look at the right triangle on the left. We are given the base angle, and since the sum of angles in a triangle is always $180^{\circ }$, the top angle must be $30^{\circ }$. Using the properties of a $30^{\circ }-60^{\circ }-90^{\circ }$ triangle, the horizontal leg on the bottom is exactly half the length of the slanted hypotenuse. Because the trapezoid is symmetric, the right triangle on the other side has identical dimensions.

Step 4: Determine the length of the bottom base

The altitudes divide the trapezoid such that the middle section forms a rectangle. Therefore, the distance between the two altitudes along the bottom base is equal to the length of the top base. We can now express the entire bottom base as the sum of its three constituent parts.

Step 5: Set up the perimeter equation

Add the four side lengths of the trapezoid together to find its perimeter in terms of our variables. We set this sum equal to the given total perimeter.

Step 6: Solve for positive integer combinations

We must find positive integer solutions for this equation. Notice that the total perimeter and the term containing $a$ are both even numbers. For the equation to balance, the term containing $c$ must also be even, meaning $c$ itself must be an even integer. Let’s test consecutive even values for $c$ to find corresponding values for $a$.

Step 7: Count the valid configurations

Each valid pair of $a$ and $c$ produces a trapezoid with a distinct set of side lengths. Because no two sets of side lengths are identical, these four trapezoids are strictly non-congruent. We stop at $c=8$ because a side length cannot be zero.

Parallelogram Decomposition Method

Drawing a line parallel to one of the slanted sides splits the trapezoid into a parallelogram and an equilateral triangle, instantly revealing the relationship between the top and bottom bases without using fractions.

Step 1: Assign variables to the unknown sides

As before, let the top base be represented by one variable and the equal slanted sides by another. We know these must be positive integers.

Step 2: Decompose the trapezoid

From the top-left vertex, draw a line down to the bottom base that is parallel to the right slanted side. Let this line intersect the bottom base at point $P$.

Step 3: Identify the parallelogram

This drawn line cuts the trapezoid into two separate shapes. On the right, the top and bottom are parallel, and the left and right are parallel by our construction. This means it is a parallelogram. In any parallelogram, opposite sides are equal.

Step 4: Analyze the remaining triangle

Now look at the triangle on the left side. We see that two of its sides have length $c$, meaning it is an isosceles triangle. Since the base angle is given as $60^{\circ }$, the other two angles must also equal $60^{\circ }$. This forces the triangle to be fully equilateral, so its third side along the base is also equal to $c$.

Step 5: Find the perimeter equation

Combine the segments along the bottom base to find its total length. Then, calculate the total perimeter of the trapezoid by summing its four outer sides. We arrive at the exact same algebraic relationship as the previous method.

Step 6: Solve for valid integers

Equate this perimeter expression to $30$. Because $c$ must be even, we test $c\in \{2,4,6,8\}$, producing four valid pairs of lengths.