Geometry Problems of the Day (Geometry Regents, June 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, June 2011

Part I: Each correct answer will receive 2 credits.


21. A man wants to place a new bird bath in his yard so that it is 30 feet from a fence, f, and also 10 feet from a light pole, P. As shown in the diagram below, the light pole is 35 feet away from the fence.

How many locations are possible for the bird bath?

1) 1
2) 2
3) 3
4) 0

Answer: 2) 2

Ten feet from the pole would be a circle. Thirty feet from the fence is a line parallel to the fence inside his yard (and, presumably, a line outside out his yard, but he can't build there).

The vertical line would cut the circle twice because 30 + 10 > 35.

Had the pole been 40 feet away, there would be 1 spot. Had the pole been more than 40 feet, there would be 0 locations.

Three locations isn't possible.

22. As shown on the graph below, △R′S′T′ is the image of △RST under a single transformation.

Which transformation does this graph represent?

1) glide reflection
2) line reflection
3) rotation
4) translation

Answer: 3) rotation

This is a 180 degree rotation about the origin.

Had it been a reflection across the line y = -x, then R' and S' would be in the other spots.

A translation would not change the orientation of the triangle.

23. Which line is parallel to the line whose equation is 4x + 3y = 7 and also passes through the point (−5,2)?

1) 4x + 3y = −26
2) 4x + 3y = −14
3) 3x + 4y = −7
4) 3x + 4y = 14

Answer: 2) 4x + 3y = −14

If a line is parallel, it has the same slope. It the equation is written in standard form, then the slope is -A/B. It should be obvious then that the lines that are parallel are Choices (1) and (2). Eliminate (3) and (4).

If the line passes through (-5,2) then that point is a solution to the equation. Plug in the values and see if you get a true statment.

4(-5) + 3(2) = -20 + 6 = -14 =/= -26

Eliminate Choice (1). The answer is Choice (2).

24. If the vertex angles of two isosceles triangles are congruent, then the triangles must be

1) acute
2) congruent
3) right
4) similar

Answer: 4) similar

If the vertex angles are congruent, then the base angles will be congruent. This makes the triangles similar.

The triangles could be congruent but they do not have to be. We know nothing about the lengths of any sides, aside from each triangle being isosceles.

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More to come. Comments and questions welcome.

More Regents problems.

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