SOLUTION: The coordinates of points A, B, and C are A(-4, 6), B(-1, 2), and C(2,-2). (a) Show that AB = BC by using the distance formula. (b) Show that AB + BC = AC by using the dist

Question 1207839: The coordinates of points A, B, and C are A(-4, 6), B(-1, 2), and C(2,-2).
(a) Show that AB = BC by using the distance formula.
(b) Show that AB + BC = AC by using the distance formula.
(c) What can you conclude from parts (a) and (b)?
Found 4 solutions by math_tutor2020, mccravyedwin, ikleyn, Edwin McCravy:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Let's find the distance from A to B.
A = (x1,y1) = (-4,6)
B = (x2,y2) = (-1,2)

The distance from A to B is exactly 5 units.
Therefore, segment AB is 5 units long.

Follow similar steps to compute the length of BC.
I'll let the student do this part.
You should get BC = 5 as the result.
This will prove AB = BC.

I'll also leave the scratch work for computing the length of AC to the student.
You should get AC = 10

This confirms that AB+BC = AC is the case since 5+5 = 10.
Since AB = BC, we have proven that B is the midpoint of AC.

Side note: The equation 4x+3y = 2 goes through points A,B, and C.

Answer by mccravyedwin(408)   (Show Source): You can put this solution on YOUR website!

It's always a good idea to draw a graph if the question is about points and lines. The red line is AB, the green line is BC, The black line is AC. I drew a little off from the others, but it's really the sum of the other two lines. The other tutor has told you enough about how you are to prove what you want with the distance formula. Edwin

Answer by ikleyn(52814)   (Show Source): You can put this solution on YOUR website!
.
The coordinates of points A, B, and C are A(-4, 6), B(-1, 2), and C(2,-2).
(a) Show that AB = BC by using the distance formula.
(b) Show that AB + BC = AC by using the distance formula.
(c) What can you conclude from parts (a) and (b)?
~~~~~~~~~~~~~~~~~~~~~

        The problem can be easily solved without the distance formula,
        which is, actually, non-necessary and excessive calculation job.

Vector AB with starting point A and ending point B is <(-1-(-4),2-6> = <3,-4>. Vector BC with starting point B and ending point C is <(2-(-1),-2-2> = <3,-4>. Thus vectors AB and BC are congruent: they represent the same vector. It implies that (a) Their lengths are the same, AB = BC (without using the distance formula). (b) AB + BC = AC (without using the distance formula). (c) The conclusion is that three points A, B and C lie on the same straight line and point C is the midpoint of segment AC.

Solved, by a simple way.

Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!

But this student probably doesn't know a vector from a specter.

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