A line segment in the standard (x, y) coordinate plane has endpoints at A(-1, 10) and B(7,… — ACT

A line segment in the standard $(x, y)$ coordinate plane has endpoints at $A(-1, 10)$ and $B(7, -2)$ . If point $M$ is the midpoint of this segment, what is the distance from $M$ to the origin $(0, 0)$ ?

Answer: 5

Answer

The distance from the midpoint

M

to the origin is 5.

To find the distance from the midpoint

M

to the origin

(0,0)

, we first determine the coordinates of

M

by taking the average of the coordinates of

A(-1, 10)

and

B(7, -2)

. This results in

M\left(\frac{-1+7}{2}, \frac{10+(-2)}{2}\right) = (3, 4)

. Next, we apply the distance formula between

M(3,4)

and the origin

(0,0)

to get

\sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9+16} = \sqrt{25} = 5

Step-by-Step Solution

Calculate the coordinates of the midpoint

M

of the segment

AB

The midpoint is

M(3, 4)

The midpoint formula is

M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

. Substituting the coordinates of

A(-1, 10)

and

B(7, -2)

gives the x-coordinate as

\frac{-1 + 7}{2} = 3

and the y-coordinate as

\frac{10 + (-2)}{2} = 4

Calculate the distance from the midpoint

M(3, 4)

to the origin

(0, 0)

The distance is 5.

The distance formula is

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

. Substituting the coordinates of

M(3, 4)

and the origin

(0, 0)

gives

d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5

Key Concept

Using the midpoint formula to find the center point of a line segment, and then using the distance formula to find the length between that point and another specified coordinate.

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