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 that the diagonals of a rhombus bisect each other at right angles. Therefore, \( RV \) and \( VT \) are segments that join the intersection point of the diagonals to the vertices of the rhombus. Since these segments are equal in length at the intersection point (let's call it point \( V \)), we have:

\[ RV = VT \]

We are given two expressions for the lengths of segments \( RV \) and \( VT \):

1. \( RV = 2x + 4 \)
2. \( VT = 8x - 8 \)

Since \( RV = VT \), we can set the two expressions equal to each other:

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

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

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

\[ 4 = 6x - 8 \]

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

\[ 12 = 6x \]

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

\[ x = 2 \]

Now that we have the value of \( x \), we substitute \( x \) back into the expression for \( VT \) to find its length:

\[ VT = 8x - 8 \]

Substituting \( x = 2 \):

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

\[ VT = 16 - 8 \]

\[ VT = 8 \]

Thus, the length of \( VT \) is \( \boxed{8} \).