SOLUTION: The vertices of a triangle are at A(-3, -2), B(2, 1) and C(6, 5). Find the length of the altitude from the vertex C and the length of side AB. Then compute for the area of the tr

Algebra ->  Linear-equations -> SOLUTION: The vertices of a triangle are at A(-3, -2), B(2, 1) and C(6, 5). Find the length of the altitude from the vertex C and the length of side AB. Then compute for the area of the tr      Log On


   

Question 1189504: The vertices of a triangle are at A(-3, -2), B(2, 1) and C(6, 5). Find the length of the altitude
from the vertex C and the length of side AB. Then compute for the area of the triangle.
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
The vertices of a triangle are at
A(-3, -2)
B(2, 1)
C(6, 5)

the length of side AB or c is equal to the distance between A and B
c=sqrt%28%282-%28-3%29%29%5E2%2B%281-%28-2%29%29%5E2%29%29
c=sqrt%2834%29
c=5.831

height to c lie on a line perpendicular to AB and passes through C
use coordinates of A and B first to find a slope of the line passing through A and B

m=%281%2B2%29%2F%282%2B3%29=3%2F5

then line is
y-y%5B1%5D=%283%2F5%29%28x-x%5B1%5D%29........use slope and (2, 1)
y-1=%283%2F5%29%28x-2%29
y=%283%2F5%29x-2%283%2F5%29%2B1
y=%283%2F5%29x+-+1%2F5

then, a line perpendicular to AB will be m=+-1%2F%283%2F5%29=-5%2F3

y-y%5B1%5D=%28-5%2F3%29%28x-x%5B1%5D%29.......use slope and (6, 5)
y-5=%28-5%2F3%29%28x-6%29
y=%28-5%2F3%29%28x-6%29%2B5
y=+-+%285+%2F3%29x%2B15

find intersection point
%283%2F5%29x+-+1%2F5=-+%285+%2F3%29x%2B15...solving it, you get
x+=+114%2F17
y=+-+%285+%2F3%29%28114%2F17%29%2B15
y+=+65%2F17

convert to decimal:
x+=+6.706, y+=+3.824

height is distance between C and intersection point

h=sqrt%28%286-6.706%29%5E2%2B%285-3.824%29%5E2%29
h=+4sqrt%282%2F17%29=1.372

Area:
if h+=+1.372 and c=5.831, the area will be
A=+%281%2F2%29%2A5.831%2A1.372
A=4.000066
A+=+4 square units




Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Use the Pythagorean Formula (aka distance formula) to find the length of AB:

AB%5E2=%285%5E2%2B3%5E2%29=25%2B9=34
AB=sqrt%2834%29

The length of the altitude to AB is the distance of the point (6,5) from the line containing AB.

Use the coordinates of A and B to determine that the equation of AB can be written as 3x-5y-1=0.

The distance of the point (p,q) from the line Ax+By+C=0 is given by the formula

abs%28%28Ap%2BBq%2BC%29%2Fsqrt%28A%5E2%2BB%5E2%29%29

abs%28%283%286%29-5%285%29-1%29%2Fsqrt%285%5E2%2B3%5E2%29%29

abs%28%2818-25-1%29%2Fsqrt%2834%29%29

8%2Fsqrt%2834%29

Then the area of the triangle is one-half base times height:

%281%2F2%29%28sqrt%2834%29%29%288%2Fsqrt%2834%29%29+=+4