Monday, October 11, 2021

Geometry Problems of the Day (Geometry Regents, August 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, August 2012

11. Question 11 was removed from the database. Everyone received credit for the question because it had no correct answer.

I remember when this happened and saw the problem. I also remember when a colleague used a photocpy of the exam in class, and I had to explain to the class why that question was flawed.

This occured before I started posting Regents questions and answers, so I no longer have a copy of the question or the exam. More information can be found at http://www.nysedregents.org/Geometry/812/in-geom812.pdf.

12.What is an equation of the circle shown in the graph below?

1) (x - 3)2 + (y - 4)2 = 25
2) (x + 3)2 + (y + 4)2 = 25
3) (x - 3)2 + (y - 4)2 = 10
4) (x + 3)2 + (y + 4)2 = 10

Answer: 2) (x + 3)2 + (y + 4)2 = 25

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.

The radius is 5, and 52 = 25, not 10. Eliminate CHoices (3) and (4).

The signs are flipped, so the correct choice is (2).

13. GAs shown in the diagram below, lines m and n are cut by transversal p.
If m∠1 = 4x + 14 and m∠2 = 8x + 10, lines m and n are parallel when x equals

1) 1
2) 6
3) 13
4) 17

If lines m and n are parallel then angles 1 and 2 are supplementary because angle 2 is supplementary to an angle that is congruent to 1. So they have a sum of 180. They are NOT equal.

4x + 14 + 8x + 10 = 180
12x + 24 = 180
12x = 156
x = 13

14. The angle formed by the radius of a circle and a tangent to that circle has a measure of

1) 45°
2) 90°
3) 135°
4) 180°

A tangent to a circle will always be perpendicular to the radius at the point where it intesects the circle.

This leads to a lot of Pythagorean Theorem/Right Triangle problems.

15. A sphere is inscribed inside a cube with edges of 6 cm. In cubic centimeters, what is the volume of the sphere, in terms of π?

1) 12π
2) 36π
3) 48π
4) 288π

If the cube has a 6 cm edge, then the sphere has a 6 cm diameter and a 3 cm radiua.

The formula for Volume is V = 4/3 π r3 = 4/3 π (3)3 = 36 π.

More to come. Comments and questions welcome.

More Regents problems.

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