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

Algebra ->  Graphs -> 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       Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
Here's a drawing:




Let's find the midpoint of AC:

x=%280%2Bb%29%2F2=b%2F2

y=%280%2Bc%29%2F2=c%2F2


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




Let's find the midpoint of BC:

x=%28a%2Bb%29%2F2

y=%280%2Bc%29%2F2=c%2F2


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





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

L=sqrt%28%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2%29


L=sqrt%28%28b%2F2-%28a%2Bb%29%2F2%29%5E2%2B%28c%2F2-c%2F2%29%5E2%29 Plug in the points of the midpoints


L=sqrt%28%28-a%2F2%29%5E2%2B0%5E2%29 Combine like terms and simplify

L=sqrt%28a%5E2%2F4%29 Square -a%2F2


L=a%2F2 Take the square root of a%5E2%2F4



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


m=%28y%5B1%5D-y%5B2%5D%29%2F%28x%5B1%5D-x%5B2%5D%29


m=%28c%2F2-c%2F2%29%2F%28b%2F2-%28a%2Bb%29%2F2%29 Plug in the given points


m=0%2F%28-a%2F2%29 Combine like terms and simplify


m=0 Divide


So the slope between the two midpoints is m=0


Remember, a slope of m=0 means that the line is horizontal. Since AB is also horizontal, the two lines are parallel.