Geometry Problems of the Day (Geometry Regents, January 2013)

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.


16. Which set of numbers could not represent the lengths of the sides of a right triangle?

1) {1, 3, SQRT(10)}
2) {2, 3, 4}
3) {3, 4, 5}
4) {8, 15, 17}

Answer: 2) {2, 3, 4}

You should know a number of Pythagorean Triples, where each number is a rational number. The smallest one containing three whole numbers is 3,4,5. This means that 2,3,4 could NOT possibly be a right triangle.

You can use Pythagorean Theorem to check, but you should have known already.

2² + 3² = 4 + 9 = 13 =/= 4²

1² + 3² = 1 + 9 = 10 = SQRT(10)²

3² + 4² = 9 + 16 = 25 = 5²

8² + 15² = 64 + 225 = 289 = 17²

17. How many points are 5 units from a line and also equidistant from two points on the line?

1) 1
2) 2
3) 3
4) 0

Answer: 2) 2

Look at the image below.

18. The equation of a circle is (x - 2)² + (y + 5)² = 32. What are the coordinates of the center of this circle and the length of its radius?

1) (-2,5) and 16
2) (2,-5) and 16
3) (-2,5) and 4 SQRT(2)
4) (2,-5) and 4 SQRT(2)

Answer: 4) (2,-5) and 4 SQRT(2)

The equation of a circle is (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.

It should be obvious that the square root of 32 is NOT 16, so eliminate choices (1) and (2).

The signs for h and k are flipped because of the minus signs, so the center is (2, -5) NOT (-2, 5).

The square root of 32 = SQRT(16) * SQRT(2) = 4 * SQRT(2)

14. The equation of a line is y = 2/3 x + 5. What is an equation of the line that is perpendicular to the given line and that passes through the point (4,2)?

1) y = 2/3 x - 2/3
2) y = 3/2 x - 4
3) y = -3/2 x + 7
4) y = -3/2 x + 8

Answer: 4) y = -3/2 x + 8

The slope of a line perpendicular to a given line with be the negative reciprocal of the slope of the given line. That means that the line we are looking for has a slope of -3/2. Eliminate Chpices (1) and (2).

The correct equation will have (4,2) as a sollution, so check that point in each equation:

y = -3/2 (4) + 7 = -6 + 7 = 1 =/= 2, not a solution

y = -3/2 (4) + 8 = -6 + 8 = 2, which is a solution

20. Consider the relationship between the two statements below.

If SQRT(16 + 9) =/= 4 + 3, then 5 =/= 4 + 3.
If SQRT(16 + 9) = 4 + 3, then 5 = 4 + 3.
These statements are

1) inverses
2) converses
3) contrapostives
4) biconditionals

Answer: 1) inverses

An inverse of a conditional has each part of the statement negated -- either a "NOT" is added or an existing "NOT" is removed.

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon. Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.