Geometry Problems of the Day (Geometry Regents, June 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. What is the equation of a line passing through the point (6,1) and parallel to the line whose equation is 3x = 2y + 4?

1) y = -2/3 x + 5
2) y = -2/3 x - 3
3) y = 3/2 x - 8
4) y = 3/2 x - 5

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

The point (6, 1) has to be a solution to the linear equation, and the linear equation has to have a slope that is equal to the slope of the line it is parallel to.

Rewrite 3x = 2y + 4 as 2y = 3x - 4 and then y = 3/2 - 2. The slope of the line is 3/2, so eliminate Choices (1) and (2). Those lines are perpendicular to the given line.

Substitute x = 6 into the other equations, 3/2 (6) - 8 = 9 - 8 = 1, which is the y-coordinate we are looking for. Choice (3) is the answer.

Check: 3/2 (6) - 5 = 9 - 5 = 4, which is not the point we are looking for.

17. The volume of a sphere is approximately 44.6022 cubic centimeters. What is the radius of the sphere, to the nearest tenth of a centimeter?

1) 2.2
2) 3.3
3) 4.4
4) 4.7

Answer: 1) 2.2

The Volume of a sphere, using the formula in the back of the booklet, is V = 4/3 π r³.

Write an equation using the given Volume:

4/3 π r³ = 44.6022
r³ = 44.6022 / ( (4/3)( π) )
r³ = 10.647...
r = CBRT(10.647...) = 2.199..., or about 2.2.

Check: V = (4/3)(3.141592)(2.2)³ = 44.6022...

If you don't know how to find the cube root function on your calculator, you can raise the number to the power of (1/3). Use parentheses around the fraction.

18. Points A(5,3) and B(7,6) lie on AB. Points C(6,4) and D(9,0) lie on CD. Which statement is true?

1) AB || CD
2) AB ⊥ CD
3) AB and CD are the same line.
4) AB and CD instersect, but are not perpendicular.

Answer: 4) AB and CD instersect, but are not perpendicular.

First find to two slopes to find out if either (2) or (4) are true. If the slopes are the same, check if they are the same line.

Slope of AB: (6 - 3) / (7 - 5) = 3/2

Slope of CD: (0 - 4) / (9 - 6) = -4/3

The slopes are not the same and they are not inverse reciprocials (that is, they do not have a product of -1). Therefore, they intersect but are not perpendicular.

19. Which set of equations represents two circles that have the same center?

1) x² + (y + 4)² = 16 and (x + 4)² + y² = 16
2) (x + 3)² + (y - 3)² = 16 and (x - 3)² + (y + 3)² = 25
3) (x - 7)² + (y - 2)² = 16 and (x + 7)² + (y + 2)² = 25
4) (x - 2)² + (y - 5)² = 16 and (x - 2)² + (y - 5)² = 25 angle.

Answer: 4) (x - 2)² + (y - 5)² = 16 and (x - 2)² + (y - 5)² = 25

If two circles are concentric, have the same center, then their equations will be the same excpt for the size of the radius squared, which comes are the equal sign.

Choice (1) has centers at (0, -4) and (-4, 0).

Choice (2) has centers at (-3, 3) and (3, -3).

Choice (3) has centers at (7, 2) and (-7, -2).

Choice (4) has centers at (2, 5) only.

20. Transversal EF intersects AB and CD, as shown in the diagram below.

Which statement could always be used to prove AB || CD?

1) ∠2 ≅ ∠4
2) ∠7 ≅ ∠8
3) ∠3 and ∠6 are supplementary
4) ∠1 and ∠5 are supplementary

Answer: 3) ∠3 and ∠6 are supplementary

If two lines are parallel and crossed by a transversal, then there are several statements that are true regarding corresponding, congruent and supplementary angles.

Choice (1) is true because they are vertical, but it doesn't prove anything about line CD.

Choice (2) is false. They are supplementary. Also they have nothing to do with line AB.

Choice (3) is true. Same-side interior angles will be supplementary. If they are supplementary, then the lines are parallel.

Choice (4) is false. The corresponding angles are congruent, not supplementary.

More to come. Comments and questions welcome.

More Regents problems.

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