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 2011
Part I: Each correct answer will receive 2 credits.
16. What is an equation of a circle with center (7,−3) and radius 4?
1) (x − 7)2 + (y + 3)2 = 4
2) (x + 7)2 + (y - 3)2 = 4
3) (x − 7)2 + (y + 3)2 = 16
4) (x + 7)2 + (y - 3)2 = 16
Answer: 3) (x − 7)2 + (y + 3)2 = 16
The equation of a circle is given by the equation (x − h)2 + (y - k)2 = 42, where (h,k) is the center of the circle and r is the radius.
In this example, h is 7, k is -3 and r is 4.
That means that the equation is (x − 7)2 + (y + 3)2 = 16, which is Choice (3).
17. What is the volume, in cubic centimeters, of a cylinder that has a
height of 15 cm and a diameter of 12 cm?
1) 180π
2) 540π
3) 675π
4) 2,160π
Answer: 2) 540π
The Volume of a cylinder is V = π r2 h.
Substitute what you know, and evaluate. You are given diameter, not radius. Divide the diameter (12) by 2 to get the radius (6).
So V = π (6)2 (15) = 540π, which is Choice (2).
18. Which compound statement is true?
1) A triangle has three sides and a quadrilateral has five sides.
2) A triangle has three sides if and only if a quadrilateral has five sides.
3) If a triangle has three sides, then a quadrilateral has five sides.
4) A triangle has three sides or a quadrilateral has five sides.
Answer: 4) A triangle has three sides or a quadrilateral has five sides.
Look at each of these one at a time.
Choice (1) is Flase because the "and" requires that both statements be true. However, a quadrilateral does not have five sides.
Choice (2) is False because the "if and only if" requires that both statements are True OR both statements are False. One is True but the other is False.
Choice (3) is False because in any If ... Then construction, if the first statement is True then the second statement must be True.
Choice (4) is True because the "or" only requires that one of the two statements be True.
19. The two lines represented by the equations below are graphed
on a coordinate plane.
3(x − 2) = −y − 4
Which statement best describes the two lines?
1) The lines are parallel
2) The lines are the same line.
3) The lines are perpendicular.
4) The lines intersect at an angle other than 90°.
Answer: 4) The lines intersect at an angle other than 90°.
Parallel lines have the same slope but different y-intercepts. If the y-intercepts are also the same, then they are the same line. Perpendicular lines have slopes that are inverse reciprocals (having a product of -1). If all other cases, they intersect, but not at right angles.
Start by finding the slopes of the two lines.
The first line is in Standard Form, so it's slope is -A/B. If you forgot that, you could rewrite the equation in slope-intercept form. The slope is -1/6.
The second line can be manipulated into slope-intercept form by adding 4 to both sides and multiplying the equation by -1.
That yields y = -3(x - 2) - 4, which has a slope of -3.
The product of 1/6 and -3 is not -1, and the slopes are obviously not the same. So the answer is Choice (4).
20. Which diagram shows the construction of the perpendicular bisector
of AB ?
Answer: 1) [See Image]
To construct a perpendicular bisector of a line segment, put the compass on one endpoint and open it up so that it is wider than half of the segment. Make an arc. Go to the other point and do the same thing without changing the compass at all.
Choice (1) shows this construction.
Choice (2) is close, but the arcs aren't big enough. They intersect on the line segment, so there aren't two points to connect to draw the line.
Choice (3) and (4) show random arcs, neither of which provide a second point to draw a line. (Two of the arcs in (4) are correct, but there need to be a matching pair below the line.)
More to come. Comments and questions welcome.
More Regents problems.
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