Question
Use the image to answer the question.
A coordinate grid shows the x-axis ranging from negative 12 to 12 and the y-axis also ranging from negative 12 to 12. Points labeled with winter activities are plotted. Sledding is plotted at left parenthesis 2 comma 12 right parenthesis. Skiing is plotted at left parenthesis negative 8 comma negative 4 right parenthesis, and ice skating is plotted at left parenthesis 2 comma negative 4 right parenthesis.
Shahriyar is at a winter park with his family. They go sledding first, then ice skating, then skiing. What is the total distance traveled between these locations?
(1 point)
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To calculate the total distance Shahriyar's family travels between the locations, we must use the coordinates given for each activity to determine the distances between the points on the grid. The distance between two points (x1, y1) and (x2, y2) in a coordinate grid can be calculated using the distance formula:
$$
\text{distance} = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}
$$
From the provided coordinates:
Sledding: (2, 12)
Ice Skating: (2, -4)
Skiing: (-8, -4)
First, we'll calculate the distance from Sledding to Ice Skating:
Both activities have the same x-coordinate (2), so we can calculate the distance just by looking at the change in the y-coordinate.
$$
\text{distance}_{\text{sledding to ice skating}} = |12 - (-4)| = |12 + 4| = 16
$$
Next, we calculate the distance from Ice Skating to Skiing:
Since the x-coordinates are different, we use the distance formula.
$$
\text{distance}_{\text{ice skating to skiing}} = \sqrt{(-8 - 2)^2 + (-4 - (-4))^2 }
$$
$$
= \sqrt{(-10)^2 + (0)^2}
$$
$$
= \sqrt{100 + 0}
$$
$$
= \sqrt{100}
$$
$$
= 10
$$
Now we add the two distances together to get the total distance:
$$
\text{total distance} = \text{distance}_{\text{sledding to ice skating}} + \text{distance}_{\text{ice skating to skiing}}
$$
$$
= 16 + 10 = 26
$$
The total distance traveled between these locations (Sledding, Ice Skating, and Skiing) is 26 units on the grid.