In the diagram below, quadrilateral TURS is a rhombus. If RV = 2x+4 and VT = 8x-8, find the length of VT. You must show all steps in order to receive full credit. You may upload a picture of your work by using the icon or your may type out your work.

Answer 1

To find the length of VT in the rhombus TURS, we know a few properties of rhombuses that can help us.

1. In a rhombus, the diagonals bisect each other at right angles.
2. The diagonals of a rhombus split the rhombus into four congruent right triangles.

Given:
RV = 2x + 4
VT = 8x - 8

Since diagonals bisect each other in rhombus TURS, we have:

\[ RV = VT \]

This leads us to set up the equation:

\[ 2x + 4 = 8x - 8 \]

Now, we will solve for \(x\):

1. Subtract \(2x\) from both sides:

\[ 4 = 6x - 8 \]

2. Add \(8\) to both sides:

\[ 4 + 8 = 6x \]

\[ 12 = 6x \]

3. Divide both sides by \(6\):

\[ x = 2 \]

Now that we have the value of \(x\), we can find the lengths of \(RV\) and \(VT\).

To find \(VT\):

\[ VT = 8x - 8 \]

Substituting \(x = 2\):

\[ VT = 8(2) - 8 \]

\[ VT = 16 - 8 \]

\[ VT = 8 \]

Thus, the length of VT is 8.

You can verify by finding \(RV\) as well:

\[ RV = 2x + 4 \]

Substituting \(x = 2\):

\[ RV = 2(2) + 4 = 4 + 4 = 8 \]

Both segments are equal (VT = RV = 8), confirming our solution was correct.

Therefore, the length of VT is 8.