Questions: The midpoint M of ST has coordinates (-47,40.5). Point S has coordinates (-87,97). Find the coordinates of point T. Write the coordinates as decimals or integers. T= ( )

The midpoint M of ST has coordinates (-47,40.5). Point S has coordinates (-87,97). Find the coordinates of point T.

Write the coordinates as decimals or integers.
T= ( )

Transcript text: The midpoint $M$ of $\overline{S T}$ has coordinates $(-47,40.5)$. Point $S$ has coordinates $(-87,97)$. Find the coordinates of point $T$. Write the coordinates as decimals or integers. $T=($ $\square$ $\square$ Submit

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 coordinates of $ M $ and $ S $, we can set up equations to solve for $ x_2 $ and $ y_2 $.

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 $ of the line segment $\overline{ST}$ with coordinates $ M(-47, 40.5) $ and one endpoint $ S(-87, 97) $, 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) $ and $ T(x_2, y_2) $.

Step 2: Solve for $ x_2 $

Using the $ x $-coordinate of the midpoint: \[ -47 = \frac{-87 + x_2}{2} \] Multiply both sides by 2: \[ -94 = -87 + x_2 \] Solve for $ x_2 $: \[ x_2 = -94 + 87 \] \[ x_2 = -7 \]

Step 3: Solve for $ y_2 $

Using the $ y $-coordinate of the midpoint: \[ 40.5 = \frac{97 + y_2}{2} \] Multiply both sides by 2: \[ 81 = 97 + y_2 \] Solve for $ y_2 $: \[ y_2 = 81 - 97 \] \[ y_2 = -16 \]