find the coordinates of the point 3/10 of the way from a to b

Question

the coordinates of b is (9,7)

the coordinates of a is (-4,-6)

Answer 1

The two points we're working with are a = (-4,-6) and b = (9,7).

Remember that any line segment can be turned into the hypotenuse of a right triangle. If we make our line segment a hypotenuse, then we have one leg parallel to the x-axis that goes from a (-4,-6) to the point (9,-6), and one leg parallel to the y-axis that goes from that same point (9,-6), up to b (9,7).

Subtracting the first x-value from the second gives us the length of the bottom leg: 9 - (-4) = 13.

Subtracting the first y-value from the second gives us the length of the side leg: 7 - (-6) = 13.

As a side comment, since both legs are the same, this makes our triangle an isosceles triangle. This is because the slope of our line is 1, so our rise and run are the same.

Now that we have the legs that make up our triangle, we can multiply the length of each leg by 3/10, or 0.3, in order to get a triangle with a hypotenuse 3/10 the length of the original segment. Since both legs are the same length, both the legs of our new triangle will be 13*(3/10) = 39/10 = 3.9 units in length.

If our starting point, a, is at (-4,-6), all we have to do is add 3.9 to -4 and -6 to find the coordinates of the point 3/10 of the way from a to b.

-4 + 3.9 = -0.1, -6 + 3.9 = -2.1

So, our point has coordinates (-0.1, -2.1).

Answer 2

x-coordinate = -4 + (3/10)(change in x from a to b) = -4 + (3/10)(13) = -0.1

y-coordinate = -6 + (3/10)(change in y from a to b) = -6 + (3/10)(13) = -2.1