In the standard (x, y) coordinate plane, point P(2, -3) is an endpoint of line segment PQ,… — ACT

In the standard $(x, y)$ coordinate plane, point $P(2, -3)$ is an endpoint of line segment $PQ$ , and $M(-1, 5)$ is the midpoint of $PQ$ . What is the distance from endpoint $Q$ to the point $(0, 10)$ ?

A
2
B
$\sqrt{7}$
5Answer
D
$\sqrt{26}$
E
7

Answer

To solve this problem, first determine the coordinates of endpoint

Q(x, y)

using the midpoint formula. Since

M(-1, 5)

is the midpoint of segment

PQ

and

P(2, -3)

is one endpoint, the horizontal coordinate satisfies

-1 = \frac{2 + x}{2}

, which yields

x = -4

. The vertical coordinate satisfies

5 = \frac{-3 + y}{2}

, which yields

y = 13

. Thus, the coordinates of

Q

are

(-4, 13)

. Next, calculate the distance between

Q(-4, 13)

and the target point

(0, 10)

using the distance formula:

d = \sqrt{(0 - (-4))^2 + (10 - 13)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = 5

. This matches the correct value.

Step-by-Step Solution

Set up equations using the midpoint formula to determine the coordinates of endpoint

Q(x_Q, y_Q)

-1 = \frac{2 + x_Q}{2}

and

5 = \frac{-3 + y_Q}{2}

The coordinates of a midpoint are the averages of the coordinates of the two endpoints.

Solve the midpoint equations for the coordinates of

Q

x_Q = -4

and

y_Q = 13

, so

Q = (-4, 13)

Multiplying by two and subtracting the coordinates of the first endpoint isolates the variables.

Apply the distance formula to find the distance between

Q(-4, 13)

and the target point

(0, 10)

d = \sqrt{(0 - (-4))^2 + (10 - 13)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5

The distance formula calculates the straight line distance between two points in a coordinate plane.

Key Concept

Using the midpoint formula to find a missing endpoint and applying the distance formula to find the distance between two coordinate points.

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic