find the area of the polygon whose vertices are at (1,-4),(4,-1),(4,5),(-1,4) and (-2,-1).?

Question

this question is an assignment given to us by our teacher in analytic geometry...

Answer 1

37.5 sq. units.

I drew a picture on a coordinate plane.
I got

two 1 x 5 right triangles
one 5 x 5 square
one isocoles triangle h = 3 b = 5

added them up and here it is

37.5 sq un.

Answer 2

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Answer 3

One method for doing this would be: For each side, find its length and its perpendicular distance from the origin. Then you have the value of base and height to find the area of the triangle with that side as base and the origin as vertex. Add up the areas of the five triangles so found. However, have you done all the stuff needed for that? Equation of a line through two points, perpendicular distance of a line from the origin?
[In case that's the expected method, I've done one of them below as an example]

Simpler, though, is to divide this polygon into a trapezium and two triangles. You need to draw the sketch to follow my description:
The points (1, -4) (4, -1) (-2, -1) form a triangle with (horizontal) base 6 units and height 3 units, and so its area is 9 sq unit.
The vertical line through (4, 5) and the horizontal line through (-1, 4), together with the join of those two points, form a rightangled triangle with base 5 units and height 1 unit, so you can find its area.
What remains is a trapezium with horizontal sides 5 units and 6 units, and the vertical distance between them 5 units, so you can find its area.

Adding these three areas together gives
0.5*6*3 + 0.5*5*1 +0.5*11*5
= 39 sq unit


The other method:

The line through (-1, 4) and (4, 5) has equation
x - 5y + 21 = 0
Its perpendicular distance from the origin is
21 / sqrt(1^2 + (-5)^2)
= 21 / sqrt(26)
The distance between those points is
sqrt(5^2 + 1^2) = sqrt(26)

Hence the area of the triangle with vertex at the origin and base the join of these points is
0.5*sqrt(26)*21/sqrt(26)
= 10.5
I'm sure we could work out a simple formula for this rather than going through the whole process for each of the other four points.
Line through (x1, y1) and (x2, y2) is
(y-y1)(x2-x1) -(x-x1)(y2-y1)=0

x(y2-y1) -y(x2-x1) - x1(y2-y1) +y1(x2-x1) = 0

so the area of the triangle is
0.5*|x1y2 - x2y1| which gives 10.5 for the points we just did.
The others are
0.5*(-1+16 + 20+4 +1+8 +8+1)
=28.5, and adding 10.5 gives 39 as before.

PS I see that No0bz already knew that formula which I've just worked out.

Answer 4

I got 39. I used the same solution as the first poster, but the bottom triangle has base 6 instead of 5, so its area is .5(6)(3) = 9. The 2 right triangles each have area 2.5, and the square has area 25.

Answer 5

look its easy i wont do it but ill teach you how
its like this; graph it first, then from a counter clock wise direction, do this(dont forget to repeat the firstxy)

X1 X2 X3 X4 X5 X1
Y1 Y2 Y3 Y4 Y5 Y1

then multiply 1/2(x1y2+x2y3+x3y4+x4y5+x5y1- y1x2-y2x3-y3x4-y4x5-y5x1)

its 1/2 times summation of all diagonal down minus all diagonal up


yeah baby! haha i like my formulas easy and my solutions short...
but its nice you worked it out the long way, hy
for verification purposes..