Question

The vertices of triangle ABC are A(-2,3), B(3,5), C(0,-6). D is the midpoint of AB and E is the midpoint of BC. Show that DE is equal to half of AC.

A. DE = 1/2 AC is proven.
B. DE = AC is proven.
C. DE = 2 AC is proven.
D. DE = 1/4 AC is proven.

Answer 1

DE is equal to half of AC can be proven by finding the lengths of DE and AC and comparing them.

Step-by-step explanation:

To show that DE is equal to half of AC, we need to find the lengths of DE and AC and compare them.

First, let's find the coordinates of D and E. The midpoint formula gives us:

D = ((-2+3)/2, (3+5)/2) = (0.5, 4)

E = ((3+0)/2, (5-6)/2) = (1.5, -0.5)

Next, we can use the distance formula to find the lengths of DE and AC:

DE = sqrt((1.5-0.5)^2 + (-0.5-4)^2)

= sqrt(1^2 + 4.5^2)

= sqrt(1 + 20.25)

= sqrt(21.25)

AC = sqrt((-2-0)^2 + (3+6)^2)

= sqrt((-2)^2 + 9^2)

= sqrt(4 + 81)

= sqrt(85)

Now we can compare the two distances:

DE = sqrt(21.25)

AC = sqrt(85)

Since DE is equal to sqrt(21.25) and AC is equal to sqrt(85), DE is equal to half of AC.