Question

the units of the subway map below are in miles. suppose the routes between stations are straight. find the distance a passenger would travel between stations h and m.
the distance from station h to station m is approximately (square) miles.
(round to the nearest tenth as needed.)

Explanation:

Step1: Identify coordinates of H and M

Station H: $(2, -1)$; Station M: $(-6, 2)$

Step2: Apply distance formula

The distance formula is $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
Substitute values: $d=\sqrt{(-6-2)^2+(2-(-1))^2}$

Step3: Calculate differences

$(-6-2)=-8$, $(2-(-1))=3$
Expression: $d=\sqrt{(-8)^2+3^2}$

Step4: Compute squares and sum

$(-8)^2=64$, $3^2=9$, sum: $64+9=73$
Expression: $d=\sqrt{73}$

Step5: Evaluate and round

$\sqrt{73}\approx8.544$, wait correction: Wait no, recheck coordinates:
Wait correction Step1: Recheck graph: Station H is $(2, -1)$, Station M is $(-6, 2)$? No, wait red line is y=1? Wait no, red line is y=1? Wait no, point A is (0,0), green line passes A(0,0), E(-2,3). Blue line passes A(0,0), H(2,-1), J(4,-2), M(-6,3). Yes, M is (-6,3), H is (2,-1).
Correct Step2: $d=\sqrt{(-6-2)^2+(3-(-1))^2}$
Step3: $(-6-2)=-8$, $(3-(-1))=4$
Step4: $(-8)^2=64$, $4^2=16$, sum: $64+16=80$
Step5: $d=\sqrt{80}=4\sqrt{5}\approx8.944$? No, wait no, red line is y=1: Point C is (4,1), B(2,1), D(-4,1), M(-6,1)? Wait no, the red line is horizontal at y=1, blue line passes H(2,-1), J(4,-2), A(0,0), K(-2,1), M(-6,3). Yes, M is (-6,3), H is (2,-1).
Wait $\sqrt{(-6-2)^2+(3-(-1))^2}=\sqrt{(-8)^2+(4)^2}=\sqrt{64+16}=\sqrt{80}\approx8.9$? No, wait 8^2+4^2=64+16=80, sqrt(80)=8.944≈8.9? No, wait maybe I misread H: H is (2,1)? No, red line is y=1, blue line crosses red line at K(-2,1), H is on blue line below red line: H(2,-1). M is on blue line at (-6,3).
Wait another way: Count grid units. Horizontal distance between H(2,-1) and M(-6,3) is $|2 - (-6)|=8$ units, vertical distance is $|-1 - 3|=4$ units.
Distance is $\sqrt{8^2+4^2}=\sqrt{64+16}=\sqrt{80}=4\sqrt{5}\approx8.9$? No, wait 8.944 rounded to nearest tenth is 8.9? But wait maybe coordinates are H(3,-1), M(-7,3)? No, the grid: each square is 1 unit. A is (0,0), E is (-2,3), so green line is y=-1.5x. Blue line is y=-0.5x. So H is (2,-1) (since y=-0.52=-1), M is (-6,3) (y=-0.5(-6)=3). Yes.
Wait $\sqrt{80}\approx8.9$? No, 8.944 is 8.9 when rounded to nearest tenth. But wait maybe I misread M: M is (-6,2)? No, blue line at x=-6 is y=3.
Wait wait, maybe the red line is y=1, so D(-4,1), K(-2,1), A(0,0), H(2,-1), J(4,-2). M is (-6,3). Yes. So distance is $\sqrt{(2 - (-6))^2+(-1 - 3)^2}=\sqrt{8^2+(-4)^2}=\sqrt{64+16}=\sqrt{80}\approx8.9$ miles.

Wait no, wait I made a mistake: vertical distance is $3 - (-1)=4$, horizontal is $-6-2=-8$, absolute values. So $\sqrt{8^2+4^2}=\sqrt{80}\approx8.9$ miles.

Wait but maybe the coordinates are H(2,1)? No, H is below the red line (y=1), so y=-1. Yes.

Wait another check: $\sqrt{80}=8.94427191$, rounded to nearest tenth is 8.9.

Wait but maybe I misread the points: Let's re-express:
Station H: (2, -1)
Station M: (-6, 3)
Distance formula:

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

$$ d = \sqrt{(-6 - 2)^2 + (3 - (-1))^2} $$

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

$$ d = \sqrt{64 + 16} $$

$$ d = \sqrt{80} \approx 8.9 $$

Wait but maybe the red line is y=0? No, point A is (0,0), red line is horizontal at y=1, passing through C(4,1), B(2,1), D(-4,1). Yes, so H is (2,-1), below the red line. M is (-6,3), above the red line on the blue line.

Yes, so the final answer is approximately 8.9 miles.

Wait no, wait 8.944 is 8.9 when rounded to nearest tenth. Yes.

Correct Answer:

8.9 miles

Step1: Identify station coordinates

Station H: $(2, -1)$; Station M: $(-6, 3)$

Step2: Apply distance formula

$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
Substitute values: $d=\sqrt{(-6-2)^2+(3-(-1))^2}$

Step3: Compute coordinate differences

$(-6-2)=-8$, $(3-(-1))=4$
Expression: $d=\sqrt{(-8)^2+4^2}$

Step4: Calculate squared terms

$(-8)^2=64$, $4^2=16$; Sum: $64+16=80$
Expression: $d=\sqrt{80}$

Step5: Evaluate and round

$\sqrt{80}\approx8.9$

Answer:

11.2 miles