the coordinates of the midpoints of a quadrilateral are R(-1,3), S(3,3), T(5,-1), and U(-2,-1). Find the perimeter and the quadrilateral. Round to the nearest tenth

To find the perimeter of the quadrilateral whose midpoints are given, we can use the distance formula to calculate the length of each side of the quadrilateral and then sum them up.

### Step 1: Calculate the lengths of the sides
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

We have the midpoints \(R(-1, 3)\), \(S(3, 3)\), \(T(5, -1)\), and \(U(-2, -1)\).

#### Length of \(RS\):
\[
RS = \sqrt{(3 - (-1))^2 + (3 - 3)^2} = \sqrt{(3 + 1)^2 + 0^2} = \sqrt{4^2} = 4
\]

#### Length of \(ST\):
\[
ST = \sqrt{(5 - 3)^2 + (-1 - 3)^2} = \sqrt{2^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} \approx 4.5
\]

#### Length of \(TU\):
\[
TU = \sqrt{(-2 - 5)^2 + (-1 - (-1))^2} = \sqrt{(-7)^2 + 0^2} = \sqrt{49} = 7
\]

#### Length of \(UR\):
\[
UR = \sqrt{(-1 - (-2))^2 + (3 - (-1))^2} = \sqrt{1^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17} \approx 4.1
\]

### Step 2: Calculate the perimeter
The perimeter \(P\) of the quadrilateral is the sum of the lengths of its sides:

\[
P = RS + ST + TU + UR = 4 + 4.5 + 7 + 4.1 = 19.6
\]

### Final Answer
The perimeter of the quadrilateral is approximately **19.6** units.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How do you find the area of a quadrilateral given its vertices?
2. What is the significance of a quadrilateral’s centroid?
3. How can you determine if a quadrilateral is cyclic?
4. What is the formula for the perimeter of a parallelogram?
5. How can you find the length of a diagonal in a quadrilateral?

**Tip:** When working with distances and perimeters, always double-check your calculations for accuracy, especially when rounding.

Learn how to calculate the perimeter of a quadrilateral using midpoints R(-1,3), S(3,3), T(5,-1), and U(-2,-1). Step-by-step solution using the distance formula to find each side length and then summing them up to find the perimeter. Ideal for students and learners interested in coordinate geometry and perimeter calculations.

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