RTÆhas endpoints R(º3, 8) and T(3, 6). Find the coordinates of the
midpoint, S, of RTÆ. Then use the Distance Formula to verify that RS = ST

To solve this problem, we'll follow two steps:

1. **Find the midpoint, S, of the line segment \( \overline{RT} \)**.
2. **Verify that the distances \( RS \) and \( ST \) are equal using the distance formula**.

### Step 1: Find the Midpoint

The formula for the midpoint, \( S(x, y) \), between two points \( R(x_1, y_1) \) and \( T(x_2, y_2) \) is:

\[
S\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\]

Given the points \( R(-3, 8) \) and \( T(3, 6) \), substitute the coordinates into the formula:

\[
S\left(\frac{-3 + 3}{2}, \frac{8 + 6}{2}\right) = S(0, 7)
\]

So, the midpoint \( S \) is at \( (0, 7) \).

### Step 2: Verify \( RS = ST \) using the Distance Formula

The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

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

#### Calculate \( RS \):
The points are \( R(-3, 8) \) and \( S(0, 7) \).

\[
RS = \sqrt{(0 - (-3))^2 + (7 - 8)^2} = \sqrt{(3)^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10}
\]

#### Calculate \( ST \):
The points are \( S(0, 7) \) and \( T(3, 6) \).

\[
ST = \sqrt{(3 - 0)^2 + (6 - 7)^2} = \sqrt{(3)^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10}
\]

Since \( RS = \sqrt{10} \) and \( ST = \sqrt{10} \), we have verified that \( RS = ST \).

### Conclusion:
The coordinates of the midpoint \( S \) are \( (0, 7) \), and the distances \( RS \) and \( ST \) are equal, confirming that \( RS = ST \).

---

Would you like further details on any part of the process or have any questions?

Here are 5 related questions for further practice:
1. Find the midpoint of the line segment joining \( A(2, 5) \) and \( B(-4, 3) \).
2. Verify if the midpoint of \( P(7, 1) \) and \( Q(3, -3) \) lies on the x-axis.
3. Calculate the distance between points \( C(-5, 2) \) and \( D(1, -4) \).
4. Given two points, \( E(0, 0) \) and \( F(4, 4) \), determine the length of the segment \( \overline{EF} \).
5. If \( S \) is the midpoint of \( \overline{XY} \), and \( X(1, 2) \) and \( S(4, 5) \), find the coordinates of point \( Y \).

**Tip:** Always ensure the distances between midpoint and endpoints are equal when using the distance formula for verification!

RT has endpoints R(-3, 8) and T(3, 6). Find the coordinates of the midpoint, S, of RT. Then use the Distance Formula to verify that RS = ST.

Discover how to find the midpoint S of line segment RT with endpoints R(-3, 8) and T(3, 6). This solution covers the midpoint formula and the distance formula, verifying that RS equals ST. Suitable for students in Grades 8-10.

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