Questions: S T has a midpoint at M(-10,-6.5). Point S is at (69,-71). Find the coordinates of point T. Write the coordinates as decimals or integers. T=( , )

Transcript text: $\overline{S T}$ has a midpoint at $M(-10,-6.5)$. Point $S$ is at $(69,-71)$. Find the coordinates of point $T$. Write the coordinates as decimals or integers. T=(\square, \square)

Solution

Solution Steps

To find the coordinates of point $ T $, we can use the midpoint formula. The midpoint $ M $ of a line segment with endpoints $ S(x_1, y_1) $ and $ T(x_2, y_2) $ is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Given the midpoint $ M(-10, -6.5) $ and point $ S(69, -71) $, we can set up equations to solve for $ x_2 $ and $ y_2 $ (the coordinates of point $ T $).

Solution Approach

Use the midpoint formula to set up equations for $ x_2 $ and $ y_2 $.
Solve these equations to find the coordinates of point $ T $.

Step 1: Set Up the Midpoint Formula

Given the midpoint $ M(-10, -6.5) $ and point $ S(69, -71) $, we use the midpoint formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] where $ S(x_1, y_1) = (69, -71) $ and $ T(x_2, y_2) $ are the coordinates we need to find.

Step 2: Solve for $ x_2 $

Using the $ x $-coordinate of the midpoint: \[ -10 = \frac{69 + x_2}{2} \] Solving for $ x_2 $: \[ -20 = 69 + x_2 \] \[ x_2 = -89 \]

Step 3: Solve for $ y_2 $

Using the $ y $-coordinate of the midpoint: \[ -6.5 = \frac{-71 + y_2}{2} \] Solving for $ y_2 $: \[ -13 = -71 + y_2 \] \[ y_2 = 58 \]