Wednesday, October 20, 2021

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

6. In the diagram below of ABCD, AC ≅ BD.

Using this information, it could be proven that

1) BC = AB
2) AB = CD
3) AD - BC = CD
4) AB + CD = AD

Since BC ≅ BC, it could be removed from AC and BD, which are congruent, and the remaining segments will still be congruent because of the Subtraction Property of Congruency.

In Choice (1), there is nothing to suggest that B is a midpoint of AC.

Choice (3) is an incorrect equation. AD - BC = CD PLUS AB.

Likewise, Choice (4) is an incorrect equation. AB + CD = AD MINUS BC.

7. The diameter of a sphere is 15 inches. What is the volume of the sphere, to the nearest tenth of a cubic inch?

1) 706.9
2) 1767.1
3) 2827.4
4) 14,137.2

Answer: 2) 4 is not an odd integer; true

If the diameter is 15 then the RADIUS is 7.5, and you need the radius for the formula, which is 4/3 π r3

(4/3) π (7.5)3 = 1767.14..., or 1767.1, which is Choice (2).

If you used the diameter of 15 by mistake, you would have gotten Choice (4) and wouldn't have known you got it wrong.

Choice (1) shows the SURFACE AREA of a sphere with a 15 inch diameter.

Choice (3) showns the SURFACE AREA of a sphere with a 15 inch RADIUS.

8. The diagram below shows the construction of AB through point P parallel to CD.

Which theorem justifies this method of construction?

1) If two lines in a plane are perpendicular to a transversal at different points, then the lines are parallel.
2) If two lines in a plane are cut by a transversal to form congruent corresponding angles, then the lines are parallel.
3) If two lines in a plane are cut by a transversal to form congruent alternate interior angles, then the lines are parallel.
4) If two lines in a plane are cut by a transversal to form congruent alternate exterior angles, then the lines are parallel.

Answer: 2) If two lines in a plane are cut by a transversal to form congruent corresponding angles, then the lines are parallel.

Knowing nothing else about construction, you can deduce that the answer is Choice (2) because the angles that are marked off are corresponding angles.

The angles were not constructed to be perpendicular. The are neither alternate interior nor exterior. Eliminate Choices (1), (3) and (4).

The construction shown is how you can draw a line parallel to a given line through a point, P, not on that line. The theorem about corresponding angles justifies this.

9. Parallelogram ABCD has coordinates A(1,5), B(6,3), C(3,-1), and D(-2,1). What are the coordinates of E, the intersection of diagonals AC and BD?

1) (2,2)
2) (4.5,1)
3) (3.5,2)
4) (-1,3)

THe diagonals of a parallelogram bisect each other. You need to find the midpoint of AC or of BD. Or you can check your work by finding both of them, which MUST BE the same point.

To find the midpoint, take the average of the two x-coordinates and the average of the two y-coordinates.

E = ( (1+3)/2, (5-1)/2 ) = (2,2)

E = ( (6-2)/2, (3+1)/2 ) = (2,2)

Choice (3) is the midpoint of AB. I don't see the "logic" behind the other two incorrect choices.

10. What is the equation of a circle whose center is 4 units above the origin in the coordinate plane and whose radius is 6?

1) x2 + (y - 6)2 = 16
2) (x - 6)2 + y2 = 16
3) x2 + (y - 4)2 = 36
4) (x - 4)2 + y2 = 36

Answer: 3) x2 + (y - 4)2 = 36

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 is 6, so r2 = 36. Eliminate choices (1) and (2).

Four units above the origin is the point (0,4), which is the center. That means that the correct equation is Choice (3).

Choice (4) would have been correct if the center was four units to the RIGHT of the origin, at (4,0).

More to come. Comments and questions welcome.

More Regents problems.

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