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. You may upload a picture of your work by using the icon or your may type out your work.
Answers
GPT-4o mini
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**.
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**.