SOLUTION: Hi! I was studying analytic geometry and im having a hard to thsee queastions I dont know where to begin, I just need patterns. Hope you could help me. 1. finf the length of the

Algebra ->  Triangles -> SOLUTION: Hi! I was studying analytic geometry and im having a hard to thsee queastions I dont know where to begin, I just need patterns. Hope you could help me. 1. finf the length of the      Log On


   

Question 957774: Hi! I was studying analytic geometry and im having a hard to thsee queastions I dont know where to begin, I just need patterns. Hope you could help me.
1. finf the length of the one leg of an isosceles triangle with vertices A(5,3),B(4,-3),and C(-1,2).
2.Find the area of a triangle whose verticea are (2,3), (6,9), (2,1).
lastly: From the distance from the line 5x=12y+26 to the point P(3,-5).
Thank you
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!

Use the distance formula,
D%5BAB%5D%5E2=%284-5%29%5E2%2B%28-3-3%29%5E2
D%5BAB%5D%5E2=%28-1%29%5E2%2B%28-6%29%5E2
D%5BAB%5D%5E2=1%2B36
D%5BAB%5D%5E2=37
D%5BAB%5D%5E2=sqrt%2837%29
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From the graph, you can tell that the base of the triangle is,
b=2
and the height of the triangle is
h=4
So,
A=%281%2F2%29bh
A=%281%2F2%29%282%29%284%29
A=4
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You can find the shortest distance by finding a perpendicular line going through (3,-5).
5x=12y%2B26
12y=5x-26
y=%285%2F12%29x-13%2F6
Perpendicular line have slopes that are negative reciprocals.
y-%28-5%29=-%2812%2F5%29%28x-3%29
y%2B5=-%2812%2F5%29x%2B36%2F5
y=-%2812%2F5%29x%2B11%2F5
Find the point of intersection between the two lines.
Then find the distance between the intersection point and (3,-5).
%285%2F12%29x-13%2F6=-%2812%2F5%29x%2B11%2F5
25x-130=-144x%2B132
169x=262
x=262%2F169
So then,
y=%285%2F12%29%28262%2F169%29-13%2F6
y=655%2F1014-2197%2F1014
y=-1542%2F1014
y=-257%2F169
So then the distance is,
D%5E2=%283-262%2F169%29%5E2%2B%28-5-%28-257%2F169%29%29%5E2
D%5E2=%28245%2F169%29%5E2%2B%28-588%2F169%29%5E2
D%5E2=%2860025%29%2F169%5E2%2B%28345744%29%2F169%5E2
D%5E2=%28405769%29%2F169%5E2
D%5E2=%28637%29%5E2%2F169%5E2
D=637%2F169
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