A coordinate grid is laid over a map of a city. A subway station is located at S(-1, 9) an… — ACT

A coordinate grid is laid over a map of a city. A subway station is located at $S(-1, 9)$ and a bus terminal is located at $B(7, -1)$ . A passenger transfer center is built at the midpoint of the line segment connecting the subway station and the bus terminal. What is the distance, in grid units, between the transfer center and a parking garage located at $(7, 7)$ ?

Answer: 5 grid units

Answer

The distance between the passenger transfer center and the parking garage is 5 grid units.

First, the midpoint is determined by averaging the

x

-coordinates and

y

-coordinates of the endpoints:

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

and

y = \frac{9 - 1}{2} = 4

, giving the transfer center coordinates of

(3, 4)

. Second, the distance between

(3, 4)

and

(7, 7)

is found using the distance formula:

d = \sqrt{(7 - 3)^2 + (7 - 4)^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5

Step-by-Step Solution

Calculate the coordinates of the midpoint of the line segment connecting the subway station

S(-1, 9)

and the bus terminal

B(7, -1)

The midpoint is

M(3, 4)

The transfer center is located exactly halfway between the two endpoints, which corresponds to the midpoint.

Calculate the distance between the midpoint

M(3, 4)

and the parking garage at

(7, 7)

The distance is 5.

Applying the distance formula to

M(3, 4)

and

(7, 7)

yields the straight-line distance.

Key Concept

Distance and Midpoint Formulas

Alternative Method

Instead of using the distance formula, one can draw a right triangle on the grid with vertices at

(3, 4)

(7, 7)

, and

(7, 4)

. The horizontal leg has a length of

|7 - 3| = 4

, and the vertical leg has a length of

|7 - 4| = 3

. Using the Pythagorean theorem, the hypotenuse (distance) is

\sqrt{3^2 + 4^2} = 5

Estimated Time:1m 30s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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