Questions: JK has a midpoint at M(-3,0). Point J is at (0,-17). Find the coordinates of point K. Write the coordinates as decimals or integers. K=( )

Transcript text: 1)) $\overline{J K}$ has a midpoint at $M(-3,0)$. Point $J$ is at $(0,-17)$. 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 $ given the midpoint $ M $ and one endpoint $ J $, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint $ M $ are the averages of the coordinates of the endpoints $ J $ and $ K $. Given $ M(-3, 0) $ and $ J(0, -17) $, we can set up equations to solve for the coordinates of $ K $.

Use the midpoint formula: $ M_x = \frac{J_x + K_x}{2} $ and $ M_y = \frac{J_y + K_y}{2} $.
Substitute the known values into the equations.
Solve for $ K_x $ and $ K_y $.

Step 1: Apply the Midpoint Formula

Given the midpoint $ M(-3, 0) $ and one endpoint $ J(0, -17) $, we use the midpoint formula to find the coordinates of the other endpoint $ K $. The midpoint formula is: \[ M_x = \frac{J_x + K_x}{2} \quad \text{and} \quad M_y = \frac{J_y + K_y}{2} \]

Step 2: Set Up the Equations

Substitute the known values into the midpoint formula: \[ -3 = \frac{0 + K_x}{2} \quad \text{and} \quad 0 = \frac{-17 + K_y}{2} \]

Step 3: Solve for $ K_x $

Solve the first equation for $ K_x $: \[ -3 = \frac{K_x}{2} \implies K_x = 2 \times (-3) \implies K_x = -6 \]

Step 4: Solve for $ K_y $

Solve the second equation for $ K_y $: \[ 0 = \frac{-17 + K_y}{2} \implies 0 = -8.5 + \frac{K_y}{2} \implies K_y = 2 \times 8.5 \implies K_y = 17 \]