In the standard (x, y) coordinate plane, the midpoint of a line segment with endpoints A(r… — ACT

In the standard $(x, y)$ coordinate plane, the midpoint of a line segment with endpoints $A(r, -2)$ and $B(10, s)$ is $M(6, 3)$ . What is the distance between the point $A$ and the origin $(0, 0)$ ?

$2\sqrt{2}$ Answer
B
$3\sqrt{5}$
C
$2\sqrt{17}$
D
$4$
E
$2\sqrt{41}$

Answer

The distance between the point

A

and the origin is

2\sqrt{2}

To find the coordinates of point

A(r, -2)

, we set up the midpoint formula with the coordinates of

B(10, s)

and the midpoint

M(6, 3)

. For the

x

-coordinate,

\frac{r + 10}{2} = 6 \implies r + 10 = 12 \implies r = 2

. For the

y

-coordinate,

\frac{-2 + s}{2} = 3 \implies -2 + s = 6 \implies s = 8

. Thus, point

A

(2, -2)

. The distance from

A(2, -2)

to the origin

(0, 0)

\sqrt{(2 - 0)^2 + (-2 - 0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}

Step-by-Step Solution

Use the midpoint formula to set up equations for the coordinates of the midpoint

M(6, 3)

given the endpoints

A(r, -2)

and

B(10, s)

The equations are

\frac{r + 10}{2} = 6

and

\frac{-2 + s}{2} = 3

The midpoint coordinates are the averages of the corresponding coordinates of the endpoints.

Solve these equations for

r

and

s

r + 10 = 12 \implies r = 2

, and

-2 + s = 6 \implies s = 8

This determines the coordinates of point

A

(2, -2)

and point

B

(10, 8)

Calculate the distance between point

A(2, -2)

and the origin

(0, 0)

using the distance formula.

d = \sqrt{(2 - 0)^2 + (-2 - 0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}

The distance formula is used to find the length of the segment connecting the point to the origin.

Key Concept

Finding the endpoint of a line segment using the midpoint formula, and then finding the distance between a point and the origin using the distance formula.

Alternative Method

Instead of solving for the

y

-coordinate

s

of point

B

, we only need the

x

-coordinate of

A

, which is

r

, to find the distance between

A(r, -2)

and the origin. Since

A

y

-coordinate is already given as

-2

, we only set up the equation for the

x

-coordinate of the midpoint:

\frac{r+10}{2} = 6

, which gives

r = 2

. Thus,

A

(2, -2)

, and the distance is

\sqrt{2^2 + (-2)^2} = 2\sqrt{2}

. This saves time by not calculating

s

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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