Regular octagon ABCDEFGH, with its vertices labeled in clockwise order, is drawn on a rectangular coordinate?

Question

plane. The coordinates of A are (4,0) and the coordinates of B are (0,4). If the coordinates of vertex E are (p,q), compute p-q.

Answer 1

The final answer is -4.

All of the previous answerers appear to have concluded that the length of each side of the octagon has a length of 4. But this is wrong. With vertices at (4,0) and (0,4), segment AB has a length of sqrt(4^2 + 4^2) = sqrt(16 + 16) = sqrt(32) = 4*sqrt(2). You can also look at AB as the hypotenus of a 45-45-90 right triangle, where the hypotenuse always has length equal to sqrt(2) times the length of the legs, which are congruent.

I quickly sketched the octagon, and I found that vertex E is in the upper right of the figure, at the right end of horizontal segment DE; vertex F could also be described as being in the upper right, but it is below and to the right of E. I then wrote in 4 as the horizontal and vertical dimension of all sides that are at 45 degree angles to the horizontal, and 4*sqrt(2) as the length of all horizontal and vertical sides. I found that E was at an x-coordinate of 4 + 4*sqrt(2); if someone thought the vertical and horizontal sides were 4, they'd get 4 + 4 = 8. The y-coordinate is 4 + 4*sqrt(2) + 4 = 8 + 4*sqrt(2); if someone thought the horizontal and vertical sides were 4, they'd get 4 + 4 + 4 = 12.

The location of E is (4 + 4*sqrt(2), 8 + 4*sqrt(2)), so p = 4 + 4*sqrt(2) and q = 8 + 4*sqrt(2). Therefore, p - q, the quantity you asked for but nobody gave you, is 4 + 4*sqrt(2) - (8 + 4*sqrt(2)) = 4 - 8 + 4*sqrt(2) - 4*sqrt(2) = -4. By coincidence, this is the same answer you would get if you used the incorrect values of p = 8 and q = 12, where p - q = 8 - 12 = -4.

Answer 2

Good One, Babsie!

If we coordinate A & B, vertex E (p,q) we can solve for both.
That makes p=8, & q=12. I am not that great in math, but I did figure this one out. (I like coordinates!) I hope this lets you not just know the answer, but also, know how to get it in case you need to figure out one like it in the future.

Hope it helps! :-}

Answer 3

the component to the sq. is 6 * 6, or 36 sq. gadgets. the component to the region of the sq. that's interior the fourth quadrant (x helpful, y damaging) is 4 * 2, or 8 gadgets. So the fraction of the sq. that's interior the fourth quadrant is 8/36, or 2/9. answer: 2/9

Answer 4

It's easy!
The answer is p=8 and q=12

Answer 5

p= 8
q= 12

Answer 6

p= 8
q= 12
we can get it by using distance formula