Find Missing Sides in Isosceles Trapezoid with Perimeter 4X+2Y

To solve this problem, we'll follow these steps:

• Step 1: Understand the properties of an isosceles trapezoid.
• Step 2: Use the given information to set up equations for the perimeter.
• Step 3: Solve the equations to find the missing side length.

Now, let's work through each step:

Step 1: An isosceles trapezoid has two parallel sides and two non-parallel sides that are equal in length. Here, the lengths of the parallel sides are given as $X$ and $3X$. Therefore, the non-parallel sides (legs) must each be $Y$.

Step 2: According to the problem, the perimeter of the trapezoid is $4X + 2Y$. The perimeter can also be expressed as the sum of all sides: $X + 3X + Y + Y$.

So we write the equation:
$X + 3X + Y + Y = 4X + 2Y$

Simplify the equation:
$4X + 2Y = 4X + 2Y$

This indicates that the setup is already balanced. Thus, the lengths of the missing sides (non-parallel sides) are each $Y$.

Therefore, the solution to the problem is $Y$.