SOLUTION: Show by means of angles that A=(-1,0), B=(4,6), and C=(10,1) are the vertices of an isosceles triangle.

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Question 970823: Show by means of angles that A=(-1,0), B=(4,6), and C=(10,1) are the vertices of an isosceles triangle.
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
Show by means of angles that A=(-1,0), B=(4,6), and C=(10,1) are the vertices of an isosceles triangle.
We draw the triangle to see which angles appear to be the base angles.




We would normally use distances to show that AB ≅ AC to show that ABC is
isosceles. But we are instructed not to do it that way, but to show 
instead that ∠A ≅ ∠C.

First we find the slopes of all three sides:

Slope of AB:  m%5BAB%5D=%286-0%29%2F%284-%28-1%29%29=6%2F%284%2B1%29=6%2F5  
Slope of AC:  m%5BAC%5D=%281-0%29%2F%2810-%28-1%29%29=1%2F%2810%2B1%29=1%2F11
Slope of BC:  m%5BBC%5D=%281-6%29%2F%2810-4%29=%28-5%29%2F6=-5%2F6

The base angles are acute angles so we use absolute value
to insure that the tangent is positive:



Multiply top and bottom by 55

tan%28A%29=abs%28%2866-5%29%2F%2855%2B6%29%29=61%2F71

Now we do that for ∠C:



Multiply top and bottom by 66

tan%28C%29=abs%28%286%2B55%29%2F%2866%2B5%29%29=61%2F71

So ∠A and ∠C have the same tangents and so they have the same measure
and since the base angles of ΔABC are ≅, ΔABC is isosceles.

Edwin