Find the coordinates of the point seven tenths of the way from A to B.

Question

the coordinates of b is (12,5)

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

Answer 1

First, you can project a line from A to B, which will denote the slope, m = (y_b-y_a)/(x_b-x_a)

Here, m = (5 - (-6))/(12- (-4)) = 11/16. Converting to an angle yields Θ = tan^-1 (m) = tan^-1 (11/16) = 34.5 degrees.

Note that the line segment is projected along the x-axis with slope m with initial point (x_a,y_a) = (-4,-6).

The total distance traveled along the x-axis is |x_b - x_a| = |12- (-4))| = 16 and

the total distance traveled along the y-axis is |y_b - y_a| = |5 - (-6))| = 11.

Therefore, a right triangle is formed with the hypotenuse equal to the distance between A and B (call it c)

--> a² + b² = c² or c = √( a² + b²) = √( 11² + 16²) = √( 121 + 256) = √( 377) = 19.42.

And 7/10 * 19.42 = 13.6.

Notice that the reduced triangle is proportionately reduced along x-axis and y-axis.

Therefore, the new coordinates, (x',y') = (-4 +16*(7/10), -6+11*(7/10)) = (7.2, 1.7)

Alternatively, you can use the coordinate system to solve as follows:

The new coordinates then become (x',y') = (x_a +c*cos(Θ),y_a+ c*sin(Θ)) = (-4+13.6*cos(34.5), -6+13.6*sin(34.5)) = (7.2,1.7)

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Check:

Using distance formula, D = √ (7.2-x_a)² + (1.7 - y_a)² = √(11.2)² + (7.7)² = √184.73 = 13.6 = 7/10*(19.42) = 7/10*(Distance (A -->B)

Answer 2

After drawing and measuring; I got .11/.1

Drew a line between the points. 1/16=1 scale.

The line is19/16 long.

19/16*7/10=133/160=66.5/80=33.25/40; I used a 1/40 scale.

the point is .125/.1 (more precise).