The base of an isosceles triangle has length 2x + 2y + 4z and its perimeter is 4x + 2y + 6z. What is the length of each of the equal sides of the triangle, expressed in terms of x, y and z?

Introduction / Context:
This problem is an algebraic application of the perimeter formula for an isosceles triangle. You are given symbolic expressions for the base and the perimeter and asked to find the length of each equal side in terms of x, y and z. It tests your ability to manipulate algebraic expressions and apply the basic perimeter concept for special triangles.

Given Data / Assumptions:

• The triangle is isosceles.
• Base length B = 2x + 2y + 4z.
• Perimeter P = 4x + 2y + 6z.
• Let each equal side have length S.
• Perimeter of any triangle = sum of all three sides = B + 2S.

Concept / Approach:
In an isosceles triangle, two sides are equal and the third side is the base. If we know the total perimeter and the base length, we can subtract the base from the perimeter to obtain the combined length of the two equal sides. Dividing this result by 2 gives the length of each equal side. This logic applies equally well when the lengths are given as algebraic expressions rather than numeric values.

Step-by-Step Solution:
Let the equal side length be S. Perimeter P = B + 2S. Given P = 4x + 2y + 6z and B = 2x + 2y + 4z. So 4x + 2y + 6z = (2x + 2y + 4z) + 2S. Subtract base from both sides: (4x + 2y + 6z) - (2x + 2y + 4z) = 2S. This simplifies to 2x + 0y + 2z = 2S. So 2S = 2x + 2z, hence S = x + z.

Verification / Alternative check:
You can verify by reconstructing the perimeter using S = x + z and B = 2x + 2y + 4z. Then P = B + 2S = (2x + 2y + 4z) + 2(x + z) = 2x + 2y + 4z + 2x + 2z = 4x + 2y + 6z, which matches the given perimeter exactly, confirming that S = x + z is correct.

Why Other Options Are Wrong:
x + y and x + y + z: These values do not satisfy the perimeter equation when substituted back. 2(x + y): Leads to a perimeter expression that does not match 4x + 2y + 6z. 2x + z: This mis-allocates the z term and fails the check against the given perimeter.

Common Pitfalls:
Forgetting that there are two equal sides, and using P = B + S instead of P = B + 2S. Making sign errors when subtracting algebraic expressions. Not simplifying terms correctly, especially when combining x, y and z terms.

Final Answer:
Each of the equal sides has length x + z