Questions: J K has a midpoint at M(16.5,16). Point J is at (13,18). Find the coordinates of point K. Write the coordinates as decimals or integers. K=(

Transcript text: $\overline{J K}$ has a midpoint at $M(16.5,16)$. Point $J$ is at $(13,18)$. Find the coordinates of point $K$. Write the coordinates as decimals or integers. \[ K=( \] $\square$ $\square$

Solution

Solution Steps

To find the coordinates of point $ K $, we can use the midpoint formula. The midpoint $ M $ of a line segment with endpoints $ J(x_1, y_1) $ and $ K(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(16.5, 16) $ and point $ J(13, 18) $, we can set up equations to solve for $ x_2 $ and $ y_2 $ (the coordinates of point $ K $).

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 $ K $.

Step 1: Use the Midpoint Formula

Given the midpoint $ M(16.5, 16) $ and point $ J(13, 18) $, we use the midpoint formula to find the coordinates of point $ K $. The midpoint formula is: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Step 2: Set Up Equations

We set up the equations for $ x_2 $ and $ y_2 $ using the given midpoint and point $ J $: \[ 16.5 = \frac{13 + x_2}{2} \] \[ 16 = \frac{18 + y_2}{2} \]

Step 3: Solve for $ x_2 $

Solve the equation for $ x_2 $: \[ 16.5 = \frac{13 + x_2}{2} \] \[ 33 = 13 + x_2 \] \[ x_2 = 33 - 13 \] \[ x_2 = 20.0 \]

Step 4: Solve for $ y_2 $

Solve the equation for $ y_2 $: \[ 16 = \frac{18 + y_2}{2} \] \[ 32 = 18 + y_2 \] \[ y_2 = 32 - 18 \] \[ y_2 = 14 \]