SOLUTION: 5. Position a triangle ABC on a graph so the point A is at (0, 0), point B is at (a, 0) and point C is at (b, c). Compute the midpoints of sides AC and BC. Show that the length of

Question 127647: 5. Position a triangle ABC on a graph so the point A is at (0, 0), point B is at (a, 0) and point C is at (b, c). Compute the midpoints of sides AC and BC. Show that the length of the line segment connecting these midpoints is one half of the length of side AB and show that this line segment is parallel to side AB.

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Here's a drawing:

Let's find the midpoint of AC:

So the midpoint of AC is (b/2, c/2)

Let's find the midpoint of BC:

So the midpoint of BC is (a+b/2, c/2)

Now let's compute the length of line that connects the two midpoints

Plug in the points of the midpoints

Combine like terms and simplify

Square

Take the square root of

Since we can see that the length of AB is a (ie it's "a" units from the origin), this shows us that the length of the line that connects the two midpoints is half of AB

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Now let's find the slope of the line that connects the two midpoints

Plug in the given points

Combine like terms and simplify

Divide

So the slope between the two midpoints is

Remember, a slope of means that the line is horizontal. Since AB is also horizontal, the two lines are parallel.

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