Geometry Problems of the Day (Geometry Regents, August 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, August 2013

Part I: Each correct answer will receive 2 credits.


21. Two prisms have equal heights and equal volumes. The base of one is a pentagon and the base of the other is a square. If the area of the pentagonal base is 36 square inches, how many inches are in the length of each side of the square base?

1) 6
2) 9
3) 24
4) 36

Answer: 1) 6

This entire question boils down to "A square has an Area of 36 square inches. What is the length of the side?"

The Volumes of the two objects are the same and the heights are the same, so the Areas of the two Bases are the same. The area of the pentagon is 36, so the area of the square is 36.

A = s²
36 = s²
6 = s

Choice (2) is if the Perimeter of the square was 36. Its perimeter is 24, which is Choice (3).

22.What is the difference between the sum of the measures of the interior angles of a regular pentagon and the sum of the measures of the exterior angles of a regular pentagon?

1) 36
2) 72
3) 108
4) 180

Answer: 1) 36

A pentagon has 540 interior degrees, which is 180 * 3. A pentagon (like every other polygon) has a total of 360 exterior degrees. The difference between the two is 540 - 360 = 180.

Choice (1) is the difference between one interior and one exterior angle. Divide 180 by 5 and you'll get 360.

Choice (2) is the size of the exterior angle

Choice (3) is the size of the interior angle.

23. If line ℓ is perpendicular to distinct planes P and Q, then planes P and Q

1) are parallel
2) contain line ℓ
3) are perpendicular
4) intersect, but are not perpendicular

Answer: 1) are parallel

Imagine a rectangle, or the margin running down the side of a page, or a pillar between the floor and the ceiling. If the line is perpendicular to the each plane than the planes are parallel to each other.

Neither plane can contain line ℓ because it is perpendicular, so it only touches each plane once. The planes cannot be perpendicular to a line and to each other as well because the angles don't add up. In fact, they cannot intersect at all.

Basically, the line is a transversal in three-dimensional space.

24. Which graph represents a circle whose equation is x² + (y - 2)² = 4?

Answer: 1) (see image)

Once again, the formula for the equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center of the circle, and r is the radius. Note that there are MINUS SIGNS in the formula. (I'm at the point where I'm literally cutting and pasting this sentence because of the HTML involved in typing it.)

The radius equals the SQRT(4) = 2, so eliminate Choices (2) and (4). There's a minus in the formula, so the center of the circle is (0, 2), not (0, -2).

More to come. Comments and questions welcome.

More Regents problems.

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