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.


11. What is the length of RS with R(-2,3) and S(4,5)?

1) 2 SQRT(2)
2) 40
3) 2 SQRT(10)
4) 2 SQRT(17)

Answer: 3) 2 SQRT(10)

You can use the Distance Formula, or sketch the points on graph paper and use Pythagorean Theorem, which by now you should realize are essentially the same thing in different forms.

The difference in the x-values is 4 - (-2) = 6. The difference in the y-values is 5 - 3 = 2.

The square root of (6² + 2²) = SQRT(36 + 4) = SQRT (40).

At this point, you can eliminate Choice (2) 40. Since radical 40 is not a choice, you either have to simplify the radical, or you can convert it to a decimal and convert the other choices to decimals, too.

SQRT(40) = SQRT (2 * 2 * 2 * 5) = 2 * SQRT (2 * 5) = 2 SQRT(10).

2 SQRT(2) = 2.8

2 SQRT(10) = 6.3

2 SQRT(17) = 8.2

The square root of 40 has be between 6 and 7.

12. What are the truth values of the statement “Two is prime” and its negation?

1) The statement is false and its negation is true.
2) The statement is false and its negation is false.
3) The statement is true and its negation is true
4) The statement is true and its negation is false.

Answer: 4) The statement is true and its negation is false.

A statement and its negation cannot both be true or both be false. If one is true, the other must be false, and vice versa.

Two is a prime number. The statement is True. The negation is "Two is not a prime number", which is false.

13. A regular polygon has an exterior angle that measures 45°. How many sides does the polygon have?

1) 10
2) 8
3) 6
4) 4

Answer: 2) 8

The sum of the exterior angles of a polygon is 360°. If it is a regular polygon, then all of the angles are the same size.

Divide 360/45 = 8. There are 8 exterior angles, so the polygon has 8 sides.

The longer way is the find the regular polygon with interior angles = 135°. You might have memorized the fact that that would be an octagon, or you could work it out with the formula.

The formula is (n - 2) (180) / n = 135. You can substitute the choices or solve.

(n - 2) (180) / n = 135
180n - 360 = 135n
45n - 360 = 0
45n = 360
n = 8

The point (h, k) is (-1, 3), so the correct choice is (1).

Choice (2) does not have the signs reversed.

14. In rhombus ABCD, with diagonals AC and DB, AD = 10.

If the length of diagonal AC is 12, what is the length of DB?

1) 8
2) 16
3) SQRT (44)
4) SQRT (136)

Answer: 2) 16

Call the intersection of AC and BD point E. The two diagonals form right angles, and thus create four right triangles. The two diagonals also bisect each other. So if AC = 12, then AE = 6.

You now have a right triangle with legs 6 and DE and hypotenuse 10. Using Pythagorean Theorem, or just by memorizing a very simple Pythagorean Triple, DE = 8. Therefore, DB = 16 because E is the midpoint of DB.

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.

15. If the surface area of a sphere is 144π square centimeters, what is the length of the diameter of the sphere, in centimeters?

1) 36
2) 18
3) 12
4) 6

Answer: 3) 12

The Surface Area of a sphere is S = 4 π r².

S = 4 π r² = 144 π
r² = 144 π / (4 π)
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.