Topscore Math Problem

[notsowisetooth]

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In a two dimension coordinate system, point A=(3,-6), point B (1,1) and point C=(7,6). Find the area of triangle ABC.


How would I go about this?

 

[Shinpe]

In a two dimension coordinate system, point A=(3,6), point B (1,1) and point C=(7,6). Find the area of triangle ABC.


How would I go about this?

The area of any triangle is 0.5*base*height, where height is considered the perpendicular distance from the third point to the base.
In that triangle, if you draw it out, the base should be (3,6)->(7,6) (aka, 4), and the height the perpendicular distance between (1,1) and the base (aka, 5).
so the area, I believe should be 0.5*5*4=10. Unless I'm picturing the triangle wrong in my head, but the 0.5*b*h would hold nevertheless.

 

[notsowisetooth]

woops i meant (3,-6) for A

 

[Shinpe]

woops i meant (3,-6) for A

Well the same definition applies, it's now just a longer problem.
Set any one of the three sides as the base
let's say (3,-6)and (7,6), then your base's length is: sqrt(16+144)=sqrt(160)=4*sqrt(10)

Then you need to find the perpendicular distance between the other point to this line (and use this equation http://www.youtube.com/watch?v=5N9t5qOSzCU )

Find the equation of the other line:
m=12/4=3. y=mx+b ->6=3*7+b => b=-15
y=3*x-15 or 3*x-y-15=0

Find the distance:
D=|1*3+1*(-1)-15|/sqrt(3^2+1^2)
D=13/sqrt(10)

Area = 0.5* base* height = 0.5 * 4* sqrt (10) * 13/sqrt(10) = 26

Check my work, but I think that should be the way to go.

I wonder if there's a shorter way though ..........

 

[Streetwolf]

Yeah. Two shorter ways. The first involves using the determinant:

(3, -6)
(1, 1)
(7, 6)

Okay first put one point on the origin to make this easier. I choose point (1,1). So everything goes down 1 and left 1. So we have (0, 0), (2, -7), and (6, 5).

Now we have two points and the origin. The determinant uses the two points to find the area of the parallelogram made through and beyond those points. If we take that area and divide by 2, we'll get the area of the triangle.

Written as a matrix:

2 -7
6 5

Determinant is | (a)(d) - (b)(c) | (absolute value)
That would be (2)(5) - (-7)(6) or 10 - -42 or 52.

Divide by 2 and get 26, the area.

===

Method 2 uses the law of cosines to find the area. First find the distance between each of the 3 points:

(1,1) to (7,6) = sqrt(61) = a
(1,1) to (3, -6) = sqrt(53) = b
(3, -6) to (7,6) = sqrt(160) = c

Okay now find the value of 's' = (a + b + c) / 2
Then use Area = sqrt( (s)(s-a)(s-b)(s-c) )

For this problem you should absolutely NOT use this method because you can't really do those square roots without a calculator, and you can't get through the calculations easily with the time you have. But if you got distances that were halfway decent then I would recommend using this.

 

[notsowisetooth]

awesome, thanks guys