Geometry Problems of the Day (Geometry Regents, August 2010)

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 2010

Part I: Each correct answer will receive 2 credits.


16. In which polygon does the sum of the measures of the interior angles equal the sum of the measures of the exterior angles?

1) triangle
2) hexagon
3) octagon
4) quadrilateral

Answer: 4) quadrilateral

The sum of the external angles of ANY polygon is 360 degrees. The sum of the interior angles of a polygon is the number of sides minus 2, times 180.

A triangle has 180 degrees. A quadrilateral has 360 degrees. A pentagon has 540 degrees. A hexagon has 720 degrees. A heptagon has 900 degrees. An octagon has 1080 degrees. Etc.

The correct answer is Choice (4).

17. In the diagram below of circle O, chords AB and CD intersect at E.

If CE = 10, ED = 6, and AE = 4, what is the length of EB ?

1) 15
2) 12
3) 6.7
4) 2.4

Answer: 1) 15

If two chords intersect, the products of their segments will be equal.

So (CD)(ED) = (AE)(EB)

Which means (10)(6) = (4)(EB)

EB = 60 / 4 = 15, which is Choice (1).

18. In the diagram below of △ABC, medians AD, BE, and CF intersect at G.

If CF = 24, what is the length of FG?

1) 8
2) 10
3) 12
4) 16

Answer: 1) 8

The ratio of CF:FG is 3:1. The centroid splits to the median into two segments at a ratio of 2:1.

One third of 24 is 8, which is Choice (1).

19. If a line segment has endpoints A(3x + 5, 3y) and B(x − 1, −y), what are the coordinates of the midpoint of AB?

1) (x + 3, 2y)
2) (2x + 2, y)
3) (2x + 3, y)
4) (4x + 4, 2y)

Answer: 2) (2x + 2, y)

To find the midpoint, you have to find the middle value of the x-coordinates and of the y-coordinates. To do that, add the expressions together and divide by two.

3x + 5 + x - 1 = 4x + 4. Dividing that by two gives a quotient of 2x + 2, which is Choice (2).

3y - y = 2y. Divide that by 2, and you get just y. That is also Choice (2). (And also Choice (3), which we eliminated already.)

20. If the surface area of a sphere is represented by 144π, what is the volume in terms of π?

1) 36π
2) 48π
3) 216π
4) 288π

Answer: 4) 288π

The surface area of a sphere can be found using the formula SA = 4πr².

If 4πr² = 144π then r² = 36, and r = 6.

The Volume of a sphere can be found using the formula V = (4/3)πr³.

V = (4/3)π(6)³ = V = 288π, which is Choice (4).

More to come. Comments and questions welcome.

More Regents problems.

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