An isosceles triangle ΔABC has its altitude at the y-axis and its base at the x-axis. D is the midpoint of segment AC and E is the midpoint of segment CB. Given the coordinates A(-2a, 0), B(2a, 0), and C(0, 2b), determine the coordinates of midpoints D and E.
a.)D(2a,-b); E(-a,b)
b.)D(-a,b); E(a,b)
c.)D(0,2b); E(2a,0)
To find the coordinates of the midpoints D and E, we need to find the average of the x-coordinates and y-coordinates of the two given points.
For point D, we need to find the average of the x-coordinates and y-coordinates of points A and C.
The x-coordinate of point D is the average of -2a and 0, which is (-2a + 0)/2 = -a.
The y-coordinate of point D is the average of 0 and 2b, which is (0 + 2b)/2 = b.
Therefore, the coordinates of point D are (-a, b).
For point E, we need to find the average of the x-coordinates and y-coordinates of points C and B.
The x-coordinate of point E is the average of 0 and 2a, which is (0 + 2a)/2 = a.
The y-coordinate of point E is the average of 2b and 0, which is (2b + 0)/2 = b.
Therefore, the coordinates of point E are (a, b).
Comparing the coordinates we found with the given options:
a.) D(2a,-b); E(-a,b) - The coordinates of D and E do not match.
b.) D(-a,b); E(a,b) - The coordinates of D match, but the coordinates of E do not.
c.) D(0,2b); E(2a,0) - The coordinates of E match, but the coordinates of D do not.
Therefore, the correct answer is option b.) D(-a,b); E(a,b).