Question

For context, a quadrilateral is shown, with top left being R, top right being S, bottom left being U, bottom right being T, and V being in the middle of 2 diagonal line inside the quadrilateral. on the 2 diagonal lines are 4 equations, near the R side is 3m+2, near the S side is 2n+2, near the U side is 8+n, and near the T side is 4m-3.

If quadrilateral RSTU is a parallelogram, what are the lengths of the diagonals of the figure?

Diagonal RT is equal to ____ and diagonal SU is equal to ____.

Answer 1

To find the lengths of the diagonals RT and SU in parallelogram RSTU, we need to equate the expressions associated with each pair of opposite sides from the information provided.

First, let's define the equations given for each diagonal:

- The equation for diagonal RT is \(3m + 2\) on one side and \(4m - 3\) on the opposite side.
- The equation for diagonal SU is \(2n + 2\) on one side and \(8 + n\) on the opposite side.

Since RSTU is a parallelogram, the lengths of the diagonals RT and SU are equal when measured from opposite vertices. Thus, we can set up the following equations based on the property of a parallelogram:

1. For diagonal RT:
\[
3m + 2 = 4m - 3
\]

2. For diagonal SU:
\[
2n + 2 = 8 + n
\]

Let's solve each equation:

**1. Solve for m in the RT equation:**
\[
3m + 2 = 4m - 3
\]
Subtract \(3m\) from both sides:
\[
2 = m - 3
\]
Add 3 to both sides:
\[
m = 5
\]
Now substitute \(m\) back in to find the length of diagonal RT:
\[
RT = 3m + 2 = 3(5) + 2 = 15 + 2 = 17
\]

**2. Solve for n in the SU equation:**
\[
2n + 2 = 8 + n
\]
Subtract \(n\) from both sides:
\[
2n - n + 2 = 8
\]
This simplifies to:
\[
n + 2 = 8
\]
Subtract 2 from both sides:
\[
n = 6
\]
Now substitute \(n\) back in to find the length of diagonal SU:
\[
SU = 2n + 2 = 2(6) + 2 = 12 + 2 = 14
\]

Combining these results, we find the lengths of the diagonals:
- Diagonal RT is equal to **17**.
- Diagonal SU is equal to **14**.

Thus, the final answer is:
- Diagonal RT is equal to **17** and diagonal SU is equal to **14**.