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, June 2013
Part I: Each correct answer will receive 2 credits.
6. Which equation represents the circle whose center is (-5,3) and
that passes through the point (-1,3)?
1) (x + 1)2 + (y - 3)2 = 16
2) (x - 1)2 + (y + 3)2 = 16
3) (x + 5)2 + (y - 3)2 = 16
4) (x - 5)2 + (y + 3)2 = 16
Answer: 3) (x + 5)2 + (y - 3)2 = 16
The equation of a circle is given by the formula (x - h)2 + (y - k)2 = r2, where (h, k) is the center of the circle and r is the radius. Note that there are MINUS signs in the formula, so if h or k is negative, then a plus will be indicated.
The center is (-5, 3), so eliminate choices (1) and (2) immediately. Since the x coordinate is negative there is a plus sign in the equation (x + 5). That is Choice (3).
7. As shown in the diagram below, when right triangle DAB is reflected
over the x-axis, its image is triangle DCB.
Which statement justifies why AB ≅ CB?
1) Distance is preserved under reflection.
2) Orientation is preserved under reflection.
3) Points on the line of reflection remain invariant.
4) Right angles remain congruent under reflection.
Answer: 1) Distance is preserved under reflection.
The two lines remain congruent because the distance (length) is preserved.
Choice (2) says orientation is preserved in a reflection. This is not true.
Choice (3) is a true statement but doesn't have anything to do with lines AB and CB, which are not on the line of reflection.
Choice (4) is a true statement but doesn't have anything to do with lines AB and CB.
8. In ABC, m∠A = 3x + 1, m∠B = 4x - 17, and m∠C = 5x - 20.
Which type of triangle is ABC?
1) right
2) scalene
3) isosceles
4) equilateral
Answer: 3) isosceles
You can immediately eliminate right because if the triangle is right, it still have to be one of the other three.
You have to find the size of each of the angles to determine how many are congruent, 0, 2, or 3.
3x + 1 + 4x - 17 + 5x - 20 = 180
12x - 36 = 180
12x = 216
x = 18
m∠A = 3x + 1 = 3(18) + 1 = 55, m∠B = 4x - 17 = 4(18) - 17 = 55, and m∠C = 5x - 20 = 5(18) - 20 = 70.
The triangle is isosceles.
9. What is the equation for circle O shown in the graph below?
1) (x - 3)2 + (y + 1)2 = 6
2) (x + 3)2 + (y - 1)2 = 6
3) (x - 3)2 + (y + 1)2 = 9
4) (x + 3)2 + (y - 1)2 = 9
Answer: 3) (x - 3)2 + (y + 1)2 = 9
The equation of a circle is given by the formula (x - h)2 + (y - k)2 = r2, where (h, k) is the center of the circle and r is the radius. Note that there are MINUS signs in the formula, so if h or k is negative, then a plus will be indicated.
The radius is 3, and 32 = 9. Eliminate (1) and (2).
The center is (3, -1). You FLIP the signs in the equation, so the answer is (3).
10. Point A is on line m. How many distinct planes will be perpendicular computations.
to line m and pass through point A?
1) one
2) two
3) zero
4) infinte
Answer: 1) one
While an infinitie number of planes can be perpendicular to a given line -- think of an elevator in a videogame that keeps going up or down forever -- only ONE will be perpendicular to a line at a specific point.
More to come. Comments and questions welcome.
More Regents problems.
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