Math / Geometry

QuestionQuestion 17
Find the coordinates of the missing endpoint if $B$ is the midpoint of $\overline{A C}$ . $A(1,7), B(-3,1)$
Cl $\square$ $\square$

Studdy Solution

STEP 1

1. Point $B$ is the midpoint of line segment $\overline{AC}$ .
2. The coordinates of point $A$ are $(1, 7)$ .
3. The coordinates of point $B$ are $(-3, 1)$ .

STEP 2

1. Recall the formula for the midpoint of a line segment.
2. Set up equations for the coordinates of the missing endpoint $C$ .
3. Solve the equations to find the coordinates of $C$ .

STEP 3

Recall the formula for the midpoint of a line segment:
If $B(x, y)$ is the midpoint of $\overline{AC}$ , then:
$B = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
where $A(x_1, y_1)$ and $C(x_2, y_2)$ are the endpoints.

STEP 4

Set up equations for the coordinates of the missing endpoint $C(x_2, y_2)$ :
For the x-coordinate: $-3 = \frac{1 + x_2}{2}$
For the y-coordinate: $1 = \frac{7 + y_2}{2}$

STEP 5

Solve the equations to find the coordinates of $C$ :
For the x-coordinate: $-3 = \frac{1 + x_2}{2}$ Multiply both sides by 2: $-6 = 1 + x_2$ Subtract 1 from both sides: $x_2 = -7$
For the y-coordinate: $1 = \frac{7 + y_2}{2}$ Multiply both sides by 2: $2 = 7 + y_2$ Subtract 7 from both sides: $y_2 = -5$
The coordinates of the missing endpoint $C$ are:
$C(-7, -5)$
The coordinates of the missing endpoint are:
$\boxed{(-7, -5)}$

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