Final answer:
To find the point representing 70 miles into the trip, determine the ratio of 70 to 90 and multiply the total change in x and y by this ratio. Add these values to the starting point's coordinates to find the new point, which is approximately (7.67, 11.33) after rounding.
Step-by-step explanation:
The question is about determining a specific point on a map that represents a certain distance into a trip, given the start and end points of the trip represented by a line segment. To find the point that represents 70 miles into the 90-mile trip, we first calculate the total difference in the x and y coordinates between points M(3,2) and N(9,14), which are the horizontal and vertical distances respectively. Then we determine the ratio of 70 miles to the total trip distance, which is 90 miles. Using this ratio, we multiply the total differences in x and y by this ratio to find the differences from point M to the point representing 70 miles into the trip.
First, we find the total differences in coordinates:
- Difference in x: 9 - 3 = 6
- Difference in y: 14 - 2 = 12
Next, we find the ratio of 70 miles to 90 miles, which is 7/9.
We then scale the differences:
- Scaled x difference: 6 * (7/9)
- Scaled y difference: 12 * (7/9)
Adding these scaled differences to the coordinates of point M gives us the new point:
- New x coordinate: 3 + (6 * (7/9))
- New y coordinate: 2 + (12 * (7/9))
After calculating and rounding to the nearest hundredth, we get:
- New x coordinate: 3 + 4.67 = 7.67
- New y coordinate: 2 + 9.33 = 11.33
The point that represents 70 miles into the trip is approximately (7.67, 11.33).