Geometry Problems of the Day (Geometry Regents, June 2014)

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 2014

Part I: Each correct answer will receive 2 credits.


16. Which numbers could represent the lengths of the sides of a triangle?

1) 5, 9, 14
2) 7, 7, 15
3) 1, 2, 4
4) 3, 6, 8

Answer: 4) 3, 6, 8

For three line segments to form a triangle, the sum of the two smaller sides must be greater than the length of the third side. If they are equal, the segments for a line. If they are less, then the triangle cannot be closed.

5 + 9 = 14, 14 is not greater than 14.

7 + 7 < 15

1 + 2 < 4

3 + 6 > 8

17. The equation of a line is 3y + 2x = 12. What is the slope of the line perpendicular to the given line?

1) 2/3
2) 3/2
3) -2/3
4) -3/2

Answer: 2) 3/2

The slope of a line perpendicular to a given line is the Inverse Reciprocal of the slope of the given line. The two slopes will have a product of -1. The exception is for that horizontal lines (which have Zero slope) are perpendicular to vertical lines (which have No Slope).

If you convert 3y + 2x = 12 into slope-intercept form you will have
3y = -2x + 12
y = -2/3 x + 4

The inverse reciprocal to -2/3 is +3/2.

18. In the diagram below, point K is in plane P

How many lines can be drawn through K, perpendicular to plane P ?

1) 1
2) 2
3) 0
4) an infinite number

Answer: 1) 1

Only one line can be drawn perpendicular to a plane that goes through a specific point. An infinite number of perpendicular lines can be drawn through the plane, but only one through any one given point.

19. n the diagram below, AB and CD are bases of trapezoid ABCD.

If m∠B =123 and m∠D =75, what is m∠C?

1) 57
2) 75
3) 105
4) 123

Answer: 1) 57

Angles B and C are supplementary because of the parallel lines. Angles C and D have no relationship at all because it is not an isosceles trapezoid. For one thing, it doesn't say that it is, and for another the given numbers don't add up to 180.

Since B and C are supplementary, 180 - 123 = 57 = m∠C.

Choice (2), 75, is if you thought that it was isosceles, or you were just making a lazy guess.

Choice (3), 105, is if you thought ∠C and ∠D had to be supplementary for some reason.

Choice (4), 123, is not only lazy, but really silly. Even in an isosceles trapezoid, these would not be the two congruent angles.

If you look at the incorrect choices, 1) 8 assumes that you forgot to double the answer in the last step, 3) assumes you used 12 as the hypotenuse and 10 as a leg, and 4) assumes you used 10 and 6 as legs and tried to find the hypotenuse.

20. What is the equation of a line passing through the point (4,-1) and parallel to the line whose equation is 2y - x = 8?

1) y = 1/2 x - 3
2) y = 1/2 x - 1
3) y = -2x + 7
4) y = -2x + 2

Answer: 1) y = 1/2 x - 3

Parallel lines have the same slope. Once you know the slope of the given line, you can eliminate two of the choices.

2y - x = 8
2y = x - 8
y = 1/2 x - 4

Eliminate choices (3) and (4).

Plug in (4, -1) into the two remaining choices and see which is true:

-1 = 1/2 (4) - 3 = 2 - 3 = -1, TRUE!

-1 = 1/2 (4) - 1 = 2 - 1 = 1, False.

r² = 36
r = 6

D = 2 * 6 = 12

Don't forget that you were asked for the diameter, not the radius.

More to come. Comments and questions welcome.

More Regents problems.