Question

find the value of x, y, and z, in the rhombus below.

Explanation:

Step1: Recall rhombus property

In a rhombus, all sides are equal. So, we set up equations based on side - length equalities.
Let's assume the side - length equations from the given expressions on the sides of the rhombus. If we assume \(2x + 8=y + 2\) and \(y+2 = 4y - 40\) and \(4y-40 = 8\).
First, solve \(4y-40 = 8\) for \(y\).
Add 40 to both sides of the equation:
\(4y-40 + 40=8 + 40\)
\(4y=48\)
Divide both sides by 4:
\(y=\frac{48}{4}=12\)

Step2: Solve for \(x\)

Substitute \(y = 12\) into the equation \(2x+8=y + 2\).
We get \(2x+8=12 + 2\)
\(2x+8=14\)
Subtract 8 from both sides: \(2x=14 - 8=6\)
Divide both sides by 2: \(x = 3\)

Step3: Since all sides are equal, \(z\) is equal to the length of any side

Since \(4y-40 = 8\), and all sides of a rhombus are equal, \(z = 8\)

Answer:

\(x = 3\), \(y = 12\), \(z = 8\)