2 math questions I need help on please?

Question

1. Triangle A B C is drawn with vertex point A at (0,0), Vertex B at ( 4,6) and vertex C at (8,2) Midpoint D is on side A B. Midpoint E is on side B C. Midpoint F is on side A C. Find the coordinates of the endpoints of each midsegment.


2. Triangle A B C with perpendicular bisector D E is drawn from point D outside the triangle on side B C down to the midpoint of side A C. Segment A E measures 3x. Segment E C measures 2x + 25. Use algebra to determine if segment DE is a perpendicular bisector of segment AC given that segment AC = 75.

Answer 1

For the first one, the midpoint formula would be easiest to use to find your points. D is the midpoint of A and B, so we can start with that:

D = ((xa + xb)/2 , (ya + yb)/2)
D = ((0 + 4)/2 , (0 + 6)/2)
D = (2, 3)

The others are similar:
E = ((xb + xc)/2 , (yb + yc)/2)
E = ((4 + 8)/2 , (6 + 2)/2)
E = (6, 4)

F = ((xa + xc)/2 , (ya + yc)/2)
F = ((0 + 8)/2, (0 + 2)/2)
F = (4, 1)

For the second one, if DE bisects AC, then the two parts should be equal. Let's set them equal to see what happens:

3x = 2x + 25
x = 25

If we used that to find the measures of the sides, we can see if this really works out:

AE = 3x
AE = 3(25)
AE = 75

EC = 2x + 25
EC = 2(25) + 25
EC = 75

Now, since AE = 75 and EC = 75, then that's a good thing - we did the algebra correctly. However, since the total is supposed to be 75 and we have a total of 150, then E can't be a midpoint of AC.

Another way you can see if this works out is to add AE and EC and solve for x:

3x + 2x + 25 = 75
5x = 50
x = 10

But if x = 10, we have AE = 30 and EC = 45, which again shows that E isn't a midpoint (otherwise we'd have the same value for both).