Question

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.

Answer 1

To solve the problem, we need to start from the information given about rhombus TURS. A rhombus has some important properties, including that its diagonals bisect each other at right angles and that each diagonal divides the rhombus into two congruent triangles.

We know that:
- \( RV = 2x + 4 \)
- \( VT = 8x - 8 \)

Since RV and VT are parts of diagonal RT (which connects vertices R and T), we can use the property that the diagonals of a rhombus bisect each other. This means that:
\[
RV = VT
\]
So we can set up the equation:
\[
2x + 4 = 8x - 8
\]

Now, we'll solve for \(x\).

1. Start with the equation:
\[
2x + 4 = 8x - 8
\]

2. Subtract \(2x\) from both sides:
\[
4 = 6x - 8
\]

3. Add \(8\) to both sides:
\[
4 + 8 = 6x
\]
\[
12 = 6x
\]

4. Divide both sides by \(6\):
\[
x = 2
\]

Now that we found \(x\), we can substitute it back into the expressions for \(RV\) and \(VT\) to find their lengths.

Using \(x = 2\):

1. Calculate \(RV\):
\[
RV = 2(2) + 4 = 4 + 4 = 8
\]

2. Calculate \(VT\):
\[
VT = 8(2) - 8 = 16 - 8 = 8
\]

Since both lengths are equal, we have confirmed the values.

Finally, the length of \(VT\) is:
\[
\boxed{8}
\]