Question

find the value of ( x ), ( y ), and ( z ), in the rhombus below.
answer
( x=quad y=quad z= )

Explanation:

Step1: Recall properties of a rhombus

In a rhombus, all four sides are equal in length. So, we can set up equations by equating the expressions for the sides.
We have one side with length \(13\), another side with expression \(-2x + 9\) (wait, maybe it's a typo, but looking at the side with length \(13\) and the side with expression \(-2x + \) (maybe \(-2x + 13\)? Wait, no, the given side is \(13\) and the adjacent side with expression \(-2z + 9\) (wait, the top side is \(-2z + 9\), left side is \(13\), right side is \(2y - 5\), bottom side is \(-2x + \) (maybe a typo, but let's assume the bottom side is \(-2x + 13\) or maybe the left side is \(13\), top side is \(-2z + 9\), right side is \(2y - 5\), bottom side is \(-2x + 13\)? Wait, no, let's re - examine.

Wait, the left side is \(13\), top side is \(-2z + 9\), right side is \(2y - 5\), bottom side is \(-2x + \) (maybe a typo, but let's assume that in a rhombus, all sides are equal. So, left side (\(13\)) should equal top side (\(-2z + 9\)), left side (\(13\)) should equal right side (\(2y - 5\)), and left side (\(13\)) should equal bottom side (let's assume the bottom side is \(-2x + 13\), but maybe the bottom side is \(-2x + \) (maybe the user made a typo, but let's proceed with the given:

Left side: \(13\)

Top side: \(-2z + 9\)

Right side: \(2y - 5\)

Bottom side: Let's assume the bottom side is \(-2x + 13\) (since in a rhombus, opposite sides are equal. So left side = right side, top side = bottom side.

Wait, left side (\(13\)) = right side (\(2y - 5\)):
\(2y-5 = 13\)

Top side (\(-2z + 9\)) = left side (\(13\)):
\(-2z + 9=13\)

Bottom side (let's assume it's \(-2x + 13\)) = left side (\(13\)):
\(-2x+13 = 13\) (but maybe the bottom side is \(-2x + \) (maybe the user had a typo, but let's solve the equations we can.

Step2: Solve for \(y\)

From \(2y - 5=13\)
Add \(5\) to both sides: \(2y=13 + 5=18\)
Divide both sides by \(2\): \(y=\frac{18}{2}=9\)

Step3: Solve for \(z\)

From \(-2z+9 = 13\)
Subtract \(9\) from both sides: \(-2z=13 - 9 = 4\)
Divide both sides by \(-2\): \(z=\frac{4}{-2}=- 2\)

Step4: Solve for \(x\)

Assuming the bottom side is \(-2x + 13\) (since left side is \(13\) and in a rhombus, opposite sides are equal. So, \(13=-2x + 13\)
Subtract \(13\) from both sides: \(0=-2x\)
Divide both sides by \(-2\): \(x = 0\)

Wait, let's check:

For \(z\):
Set left side (\(13\)) equal to top side (\(-2z + 9\)):
\(13=-2z + 9\)
Subtract \(9\) from both sides: \(13-9=-2z\)
\(4=-2z\)
\(z=-2\)

For \(y\):
Set left side (\(13\)) equal to right side (\(2y - 5\)):
\(13 = 2y-5\)
Add \(5\) to both sides: \(13 + 5=2y\)
\(18 = 2y\)
\(y = 9\)

For \(x\):
Set left side (\(13\)) equal to bottom side (assuming bottom side is \(-2x+13\)):
\(13=-2x + 13\)
Subtract \(13\) from both sides: \(0=-2x\)
\(x = 0\)

Answer:

\(x = 0\), \(y = 9\), \(z=-2\)