b The x- and y-coordinates of the desired point will be 34 of the way from the x- and y-coordinates of A to the x- and y-coordinates of B, respectively.
b Let's take a look at A and B on a coordinate plane again. To get to point B from A, we need to take 6 steps to the right and 8 steps up.
We want to find a point P(x, y) on the line segment AB that is 34 of the way from point A to point B. This means that to get from A to P, we want to make 34 the steps we made to get from A to B.
As we can see, to get from A to P we need to take 4.5 steps to the right and 6 steps up. Therefore, to find the coordinates of P, we can add 4.5 to the x -coordinate of A and 6 to the y -coordinate of A.
x &= 4 + 4.5 = 8.5
y &= 1 + 6 = 7
Therefore, point P(8.5, 7) is the point on AB that is 34 of the way from A to B.
c In the same coordinate plane from Part A, we will plot point C. Let's also label ∠ CAB as θ.
Since points B and C have the same x-coordinate and AC is horizontal, we can create a right triangle ABC by connecting B and C. Additionally, since AC is horizontal and BC is vertical, their lengths are the difference between the x-coordinates and y-coordinates, respectively.
Since we know the leg's length, we can find the measure of θ by using the tangent ratio.
