Question

the points A(2,3) B(4,-1) C(-1,2) are the vertices of a triangle. find the length and perpendicular from A to BC and hence the area of ABC

Answer 1

a. Find the equation of the line BC, given two points (B and c)

Then, find the perpendicular (given the negative inverse slope from a), and the point A.

Area? Area=lengthBC*lengthperpendicular*1/2

Just finding the area, there are other ways easier.

Graph it, use Pick's Theorem http://en.wikipedia.org/wiki/Pick%27s_theorem
or
http://www.mathopenref.com/coordtrianglearea.html

or to check your answer ONLY: http://www.gottfriedville.net/mathtools/triarea.html

Answer 2

consider BC as your base:
BC = √(5^2 + (-3)^2) = √34
slope of BC = -3/5

so the slope of the perpendicular = 5/3
equation of line from A to BC , using A as the point
y-3 = (5/3)(x-2)
3y - 9 = 5x - 10
5x - 3y = 1 ---- #1

equation of BC , using C
y-2 = -(3/5)(x+1)
5y - 10 = -3x -3
3x + 5y = 7 ----#2

5 times #1 --- 25x - 15y = 5
3 times #2 --- 9x + 15y = 21
add them
34x = 26
x = 26/34 = 13/17 .... ughh
in #2
3(13/17) + 5y = 7
..
y = 16/17 , yuk, the point on BC = (13/17 , 16/17)
so length of altitude
= √(2-13/17)^ + (3-16/17)^2)
= √(441/289 + 1125/289) = √(1666/289)
= 7√34/17

so the area = (1/2) base x heigh
= (1/2) * √34 * (7/17)√34 = 7 , yeahhhh


There are of course much easier ways to find the area of a triangle if you are given the three points.

The simplest way is to list the 3 points in a column repeating the first one you listed.

2 3
4 -1
-1 2
2 3

area = (1/2) | ( sum of downproducts - sum of upproducts)|
= (1/2)|(-2+ 8 - 3 -(12 + 1 + 4)|
= (1/2)| -14 |
= 7