Question

Point P(-8, 13) lies on the directed line segment AE. The endpoints are A(-19, 6) and E(x, y). What are the coordinates of point E such that AP is 2/5 of the distance from A to E?

A) E(-13, 10)
B) E(-9, 8)
C) E(-7, 7)
D) E(-6, 6)
E) E(-4, 4)

Answer 1

To find the coordinates of point E, use the midpoint formula and the given information that AP is 2/5 of the distance from A to E. Substitute the given point P(-8, 13) into the equation, simplify, and solve for AE. Use the midpoint formula with A(-19, 6) and AE to find the coordinates of point E. The coordinates of point E are E(-13, 10).

Step-by-step explanation:

To find the coordinates of point E, we can use the midpoint formula. The midpoint of a line segment is the average of the coordinates of its endpoints. Let's use the point P(-8, 13) as one endpoint and the given information that AP is 2/5 of the distance from A to E.

First, we calculate the distance from A to E:
Distance AE = sqrt((x - (-19))^2 + (y - 6)^2).

Since AP is 2/5 of the distance from A to E, we can write the equation:
AP = (2/5) * AE.

Then, we substitute the given point P(-8, 13) into the equation:
sqrt(((-8) - (-19))^2 + (13 - 6)^2) = (2/5) * AE.

Simplifying and solving for AE, we find that AE = 125/2.

Now, we can find the coordinates of point E by using the midpoint formula with A(-19, 6) and AE = 125/2:
x-coordinate of E = (-19 + x)/2 and y-coordinate of E = (6 + y)/2.

Substitute AE = 125/2 into the midpoint formula and solve for the coordinates of E.

After performing the calculations, we find that the coordinates of point E are E(-13, 10). Therefore, the correct answer is A) E(-13, 10).