Jimmy2588

Sorry if I am asking for too many explanations...

11.
[diagram: x and y axis. with point A at (0,2) and point B at (2,0) with a line connecting the two]

In the figure above line L (not shown) is perpendicular to segment AB and bisects segment AB. which of the following points lie on line L.
A. 0,2
B. 1,3
C. 3,1
D. 3,3
E. 3,6

I got A just by eyeballing it, but it there a mathematical explanation for it?

22.

Um... I don't know how to draw the figure, but if you have the test and can explain it, please do

23.

Three lines are drawn in a plane so that there are exactly three different intersection points. Into how many nonoverlapping regions do these lines divide the plane?

Flipsta_G

11. The segment between (0,2) and (2,0) has a slope of -1. You can verify this by finding (y2-y1)/(x2-x1) = (0-2)/(2-0) = (-2)/2 = -1

Two lines are perpendicular if their slopes are negative reciprocals of eachother. Since AB has slope -1, line L bust have slope -(1)/(-1) = 1

It bisects AB, so it must go through the midpoint of it, by definition. So we find the midpoint of AB by this formula: (((x1 +X2)/2),((y1+y2)/2))
So we get ((0+2)/2), (2+0)/2) = (1,1)

So we need a line with slope 1 passing through (1,1). This is obviously the equation y = x, which passes though (0,0)

If you didn't catch that, you could do it the long way:
Slope 1, so y = 1x + b
Plug in point (1,1) to get y-intercept
1 = 1 + b
b=0

OK, now we look for what point would go on the line y = x, which is easy because the x and y have to be the same. Therefore the answer is (3,3), or D.

Flipsta_G

I don't have the book right with me here, so I can't do the middle one.

23. Draw it out, and it is pretty easy to see the answer. If you have one line, it divides the plane into 2 (1 piece on each side of the line). If you have 2 lines, it divides the plane in 4. Think of a pie. If you add in a 3rd line, it divides two of those 4 sections (from before) into 2, thus adding 2 more. The answer is 6.

optimizerdad

23. Actually, I think you'd get 7 regions. The third line drawn would create 3 new regions, not 2.

One way to visualize this is to think of a large equilateral triangle ABC, with a small equilateral triangle DEF drawn within it. Both triangles have the same center. Now extend each side of the interior triangle till the line hits the outer triangle (i.e. extend DE, EF, FD). The outer triangle will now be split into 7 distinct regions. If you like, you can now erase the outer triangle; the #distinct regions stays at 7.

Flipsta_G

Oops, I misread the question to be 1 intersection point. My bad.

Jimmy2588

Unfortunately, I cannot visualize the 3 lined one.. I get 7 regions, but 1 cannot be overlapping, so i keep getting 6.

But thank you all for these explanations. I have a feeling I might be posting every night since I am doing 2 sections a night... some of those studyhall explanations suck.

Flipsta_G

Well, for the 3 line thing, it is a bit hard to visualize.

Just draw a triangle. The only way for 3 lines to intersect at exactly 3 pts is in a triangle. Then extend the lines of each side of the triangle. You should get 3 sections for the regions directly outside the sides, 3 sections outside the angle tips, and 1 section inside the triangle. This equals 7 non-overlapping sections.

Jimmy2588

i think i DID actually get this right. I did not know exactly what they mean by overlapping. I thought the middle region (the triangle shaped one) was considered "overlapping."

So one last question for this thread, what do they mean by nonoverlapping?

Flipsta_G

I think it just means separate, divided by a line.

gcf101

Making a drawing first and taking a question in visually often helps to avoid long calculations and associated with them mistakes.

The origin - point O (0, 0).
AO = BO = 2 ==> triangle AOB is isosceles,
Perpendicular to the middle of its base AB is also a bisector of <AOB.
In other words, L is a line of symmetry for AOB.
Don't be afraid of the "s"-word; check out this cool site
http://regentsprep.org/Regents/math/...ry/Lsymmet.htm .

Line L is a bisector of the first quadrant, and its equation is
y = x,
so all the points on line L have equal coordinates of type (n, n).

Answer D (3, 3) is the correct one.

+++++++++++++++

RE: overlapping.
How many nonoverlapping triangles are formed in a square when its diagonals are drawn?
Four.
Now, how many triangles are formed in a square when its diagonals are drawn?

Flipsta_G

8.

Yea, on that problem I would have used visualization definitely, but it would be harder to explain, and he asked for the mathematical explanation.