Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.
More Regents problems.
Geometry Regents, January 2014
Part I: Each correct answer will receive 2 credits.
Which pair of edges are not coplanar?
6. A right rectangular prism is shown in the diagram below.
1) BF and CG
2) BF and DH
3) EF and CD
4) EF and BC
Answer: 4) EF and BC
Two lines are coplanar if they lie on the same plane. Think of it this way: if each line was a stick, could you attach a piece of paper to both sticks without twisting it?
BF and CG are both up and down lines. They are coplanar. Eliminate Choice (1).
BF and DH are both up and down lines. They are coplanar. Eliminate Choice (2).
EF and CD are both front to back lines. They are coplanar. Eliminate Choice (3).
EF is a front to back line, but BC is a side to side line. This is the answer.
To check: from A to the midpoint, you need to go up 6 units from -4 to 2. Add another 6 units, and the y-coordinate of the endpoint must be 8.
This would be obvious if you make a sketch on graph paper and count the boxes, which is a fine alternative if you don't want to deal with formulas and subtracting signed numbers.
7. How many points in the coordinate plane are 3 units from the origin and also equidistant from both the x-axis and the y-axis?
1) 1
2) 2
3) 3
4) 4
Answer: 4) 4
The points that are 3 units from the origin would be on a circle, centered on the origin, with a radius of three.
The points that are equidistant from the x-axis and y-axis would lie on the angle bisectors of the axes. In other words, they would be points on the two diagonal lines y = x and y = -x.
Each of those lines will cross the circle twice. So there would be four points that fit the criteria, one in each quadrant. You can make a little sketch to see it.
8. As shown below, the medians of triangle ABC intersect at D.
If the length of BE is 12, what is the length of BD?
1) 8
2) 9
3) 3
4) 4
Answer: 1) 8
The medians meet at a centroid. The point of concurrence (an intersection of more than two lines) splits each median so that one segment is twice as long as the other.
Looking at BE, BD is 2/3 of the length of BE, and DE is 1/3 of the length of BE. And BD is twice the length of DE. The segment attached to the vertex is always the 2/3, while the segment intersecting the midpoint of the other side is the 1/3.
Since BE is 12, 2/3 of 12 is 8.
Choice (4) 4 is the length of DE.
To get either 9 or 3, you would've had to have used 1/4 and 3/4, I guess.
9. The solution of the system of equations y = x2 - 2 and y = x is
1) (1,1) and (-2,-2)
2) (2,2) and (-1,-1)
3) (1,1) and (2,2)
4) (-2,-2) and (-1,-1)
Answer: 2) (2,2) and (-1,-1)
The line y = x means that the x-coordinate and the y-coordinate have to be the same. Every choice works, so we can ignore this condition for now. Check the other condition:
y = (1)2 - 2 = 1 - 2 = -1, not 1, so eliminate Choices (1) and (3).
y = (2)2 - 2 = 4 - 2 = 2. (2, 2) is a solution, but we already eliminated Choice (3), so Choice (2) is the answer.
Check: y = (-1)2 - 2 = 1 - 2 = -1. (-1, -1) is a solution.
You could also have graphed these two equations in your graphing calculator and looked at the Table of Values for (-1, -1) and (2, 2). Obviously, (1, 1) and (-2, -2) would not be there.
Finally, you could have solved it algebraically, using substitution.
x = x2 - 2
0 = x2 - x - 2
0 = (x - 2)(x + 1)
x - 2 = 0 or x + 1 = 0
x = 2 or x = -1
y = x | y = x
y = 2 | y = -1
10. Line ℓ passes through the point (5,3) and is parallel to line k whose
equation is 5x + y = 6. An equation of line ℓ is
1) y = 1/5 x + 2
2) y = -5x + 28
3) y = 1/5x - 2
4) y = -5x - 28
Answer: 2) y = -5x + 28
Parallel means that it has the same slope, so first find the slope of the given line. Once you have it, you can eliminate 2 of the choices (which, in this case, will be perpendicular to the given line).
y = -5x + 6
Eliminate Choices (1) and (3). Next, substitute (5, 3) and see if you get a true statement.
3 ?= -25 + 28
3 = 3
Choice (2) is the answer.
You could have also figured that starting at (5, 3) with a slope of -5, to get back to the y-axis (and the y-intercept), every time you subtract 1 from the x-coordinate, you have to add 5 to the y-coordinate. You have to add 5 five times, which is 3 + 25, or 28.
More to come. Comments and questions welcome.
More Regents problems.
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