Geometry Problems of the Day (Geometry Regents, June 2012)

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 2012

Part I: Each correct answer will receive 2 credits.


16. In the diagram of ABC shown below, DE || BC.

If AB = 10, AD = 8, and AE = 12, what is the length of EC?

1) 6
2) 2
3) 3
4) 15

Answer: 3) 3

Set up a proportion to find the size of AC and subtract the length of AE to get EC.

8 / 10 = 12 / AC
8 AC = 120
AC = 15

EC = AC - AE = 15 - 12 = 3

17. What is the length of AB with endpoints A(-1,0) and B(4,-3)?

1) √(6)
2) √(18)
3) √(34)
4) √(50)

Answer: 3) √(34)

Use the Distance Formula or the Pythagorean Theorem.

√( (-1 - 4)² + (0 - -3)² ) = √( (-5)² + (3)² ) = √(34)

18. The sum of the interior angles of a polygon of n sides is

1) 360
2) 360/n
3) (n - 2) • 180
4) (n - 2) • 180 / n

Answer: 3) (n - 2) • 180

This is basically a definition question. This is the formula for the sum of the interior angles -- the number of triangles you can make times 180. The number of triangles you can make is Two less than the Number of sides that the polygon has.

Choice (1) is the total of the exterior angles. It's always 360, regardless of the number of sides.

Choice (2) is the size of each exterior angle in a regular polygon.

Choice (4) is the size of each interior angle in a regular polygon.

19. What is the slope of a line perpendicular to the line whose equation is 20x - 2y = 6?

1) -10
2) -1/10
3) 10
4) 1/10

Answer: 2) -1/10

The slope of a line perpendicular to the given line will be the inverse reciprocal of the slope of the given line.

The slope of a line written in standard form, Ax + By = C, is -A/B, which in this case is -(20)/(-2) = 10.

If you need to, you could convert to slope-intercept form:

20x - 2y = 6
-2y = -20x + 6
y = 10x - 3

The slope of a line perpendicular to a line with a slope of 10 would be -1/10.

20. Which graph represents a circle whose equation is (x + 2)² + y² = 16?

Answer: 3) (see image)

The equation of a circle is given by the formula (x - h)² + (y - k)² = r², where (h,k) in the center of the circle and r is the radius. Note that there are MINUS signs in the formula, so the signs will be flipped.

Note that this comes up so often that I cut and paste the above sentence from blog post to blog post.

The radius of the circle is √(16) = 4. Eliminate Choices (1) and (4), which have radius 8 and diameter 16.

The center of the circle, according to the equation, is (-2, 0) because the sign if flipped. That's Choice (3).

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.