Should be pretty straight forward if you use the sin / cos rules for right triangles. opposite = sin(α) * hypotenuse, adjacent = cos(α) * hypotenuse.
For example, the first line from 200,0 goes down at a -50° angle, for a length of 500. If you form a right triangle, then the opposite side (y) would be sin(50°) * 500 and the adjacent side (x) would be cos(50°) * 500.
Your drawing is obviously not to scale with length or angles, so it disguises the actual angles. If you draw it to scale, the 200 line bends back in and the 100 line bends upward. Similarly, the 600 line is actually vertical and the 700 line bends backwards at about 30°.
I've attached a drawing of what the polygon actually should look like.
http://img214.imageshack.us/img214/6959/weirdpolygonce4.gif
The x value of the left yellow dot would be:
-200 - cos(60°) * 300 + sin(40°) * 200 + cos(30°) * 100
x1 = -134.839938
The y value of the left yellow dot would be:
0 - sin(60°) * 300 - cos(40°) * 200 + sin(30°) * 100
y1 = -363.01651
The x value of the right yellow dot would be:
200 + 500 * cos(50°) - 700 * sin(30°)
x2 = 171.393805
The y value of the right yellow dot would be:
0 - (500 * sin(50°) + 600 + 700 * cos(30°))
y2 = -1589.24
Question 1:
Coordinates of the yellow dots:
Left dot (-134.8399, -363.0165)
Right dot (171.3938, -1589.2400)
Question 2:
Distance between is simple to calculate using the distance formula:
D = sqrt((y2 - y1)² + (x2 - x1)²)
D = sqrt((-1589.24 + 363.0165)² + (171.3938 + 134.8399)²)
D = 1,263.8842
Question 3:
Area of the polygon.
Here you can use the shoestring algorithm to find the area of a polygon given the coordinates. Details of this method are given in the link below:
http://mathworld.wolfram.com/PolygonArea.html
If you figure out the coordinates you find:
(-134.8399, -363.0165)
(-221.4425, -413.0165)
(-350, -259.8076)
(-200, 0)
(200, 0)
(521.3938, -383.0222)
(521.3938, -983.0222)
(171.3938, -1589.2400)
To figure out the area, list your coordinates as shown above, draw a line from x1 down to y2 and take the product. Same for x2 down to y3, etc. The last string is from x8 back to y1. These are your "positive" shoestrings.
Figure the individual products and add them up.
(x1 * y2) + (x2 * y3) + (x3 * y4) + (x4 * y5) + (x5 * y6) + (x6 + y7) + (x8 * y1)
Now do the same with y1 down to x2, y2 down to x3, ... , y8 back to x1. These are your "negative" shoestrings.
(y1 * x2) + (y2 * x3) + (y3 * x4) + ... + (y8 * x1)
Subtract the negative shoestrings from the positive shoestrings then divide by 2.
When I did this I came out with an area of:
744,884.75 sq. units.
Question 4:
It's an octagon which means the interior angles should add up to (8-2) x 180° = 1080°
So far we have 750° accounted for... so the other two angles add up to 330°. Let me work on the exact values of each angle.
Again we can form a triangle. I'm going to use the angle of the lower right dot, less the 30° from vertical. Let's call that Θ
sin(Θ) = opposite/hypotenuse.
The opposite side would be the difference of the x coordinates:
= x2 - x1 = 171.3938 - (-134.8399) = 306.2337
The hypotenuse is the distance D we calculated above:
= 1263.8842
sin(Θ) = 306.2337/1263.8842
Θ = arcsin(306.2337/1,263.8842)
Θ = arcsin(0.242295694)
Θ = 14.0221°
The angle near the first dot will be:
120° + 90° + (90° - Θ)
= 300° - Θ
= 285.9779°
Adding the 30° from the vertical, the angle for the 2nd dot is:
30° + Θ
= 44.0221°.
(As a double-check, indeed they add up to 330°, so I think we are okay at this point).
2007-11-01 11:04:03
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answer #1
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answered by Puzzling 7
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"What are the coordinates of the two yellow dots?"
I think this is 100 and 700, but I'm not sure.
"What is the distance between the two yellow dots?"
Use the formula for distance - "d = a+b/2"
Add the two numbers together(100 and 700 and divide by 2)
"What is the area of the polygon?"
I think you use the formula "A= 2(pi)r".
"What are the angles of the two yellow dots, in degrees?"
I don't know this one.
2007-11-01 17:46:55
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answer #2
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answered by w00t43 3
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