Geometry Problems of the Day (Geometry Regents, January 2011)

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, January 2011

Part I: Each correct answer will receive 2 credits.


25. Which graph represents a circle with the equation (x − 3)² + (y + 1)² = 4?

Answer: 2) [See Image]

The formula for a circle is (x − h)² + (y - k)² = r², where (h,k) is the center of the circle and r is the radius.

In this example, the center is at (3, -1) and the radius is 2 (with the diameter being 4).

This matches the second graph, Choice (2).

Choices (1) and (3) have the wrong radius. Choices (3) and (4) have the wrong center.

26. The point (3,−2) is rotated 90° about the origin and then dilated by a scale factor of 4. What are the coordinates of the resulting image?

1) (-12,8)
2) (12,-8)
3) (8,12)
4) (-8,-12)

Answer: 3) (8,12)

When the direction of the rotation is not stated, the default is counterclockwise, like the numbering of the quadrants I, II, III, IV.

Point (3,-2) becomes (2,3) as it moves from IV to I, where both cooridnates are positive.

The scale factor of 4 moves the point to (8,12).

Note that all four quadrants are represented in the choices if you made a mistake with the rotation.

27. In the diagram below of △ABC, side BC is extended to point D, m∠A = x, m∠B = 2x + 15, and m∠ACD = 5x + 5.

What is m∠B?

1) 5
2) 20
3) 25
4) 55

Answer: 3) 25

According to the Exterior Angle Theorem, m∠A + m∠B = m∠ACD.

Therefore, x + 2x + 15 = 5x + 5

3x + 15 = 5x + 5

15 = 2x + 5

10 = 2x

x = 5

Now that you know what x is, you can find the size of the angle. Angle B is 2x + 15 = 2(5) + 15 = 25, which is Choice (3).

Choice (1) is x, not the angle.

28 Point P lies on line m. Point P is also included in distinct planes Q, R, S, and T. At most, how many of these planes could be perpendicular to line m?

1) 1
2) 2
3) 3
4) 4

Answer: 1) 1

Only 1 plane can be perpendicular to a line at a given point.

End of Part I.

More to come. Comments and questions welcome.

More Regents problems.

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