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.


21. Which transformation of the line x = 3 results in an image that is perpendicular to the given line?

1) r_x-axis
2) r_y-axis
3) r_{y = x}
4) r_{x = 1}

Answer: 3) r_{y = x}

Reflecting a vertical line over a diagonal line will result in a horizontal line. The points (3,0),(3,1),(3,2),(3,3) will become (0,3),(1,3),(2,3),(3,3). They will meet at (3,3) at right angles.

Choice (1) will result in the same line. Point (3,1) will reflect to (3,-1) and (3,-1) will reflect back to (3,1). The line transforms onto itself.

Choice (2) will have an image at x = -3.

Choice (4) will have an image at x = -1 because x = 1 is in the middle of x = 3 and x = -1.

22. In the diagram below of regular pentagon ABCDE, EB is drawn.

What is the measure of ∠AEB?

1) 36°
2) 54°
3) 72°
4) 108°

Answer: 1) 36°

Each answer of a regular pentagon is 720/5 = 108 degrees. This is not the answer.

Triangle AEB has 180 degrees and angle A has 108 of them. Since AE = AB, then triangle AEB is an isosceles triangle, and AEB and ABE are the base angles, which must be the same size.

180 - 108 = 72 degrees. (Still not the answer.)

72 / 2 = 36 degrees. This is the answer, which is Choice (1).

Choice (2) seems to be if you took the 108 and divided by 2 without subtracting from 180 first.

23. △ABC is similar to △DEF. The ratio of the length of AB to the length of DE is 3:1. Which ratio is also equal to 3:1?

1) m∠A / m∠D
2) m∠B / m∠F
3) area of △ABC / area of △DEF
4) perimeter of △ABC / perimeter of △DEF

Answer: 4) perimeter of △ABC / perimeter of △DEF

A ration of 3:1 means that each side of ABC is three times the size of the corresponding sides of DEF. The corresponding angles are congruent because the shapes are similar. (How could they be 3:1? Both triangles are 180 degrees!!)

If the sides are 3 times as long that means that the SUM of the sides (the perimeter) is three times as big. This is Choice (4).

Note that if the base and the altitude of ABC is 3 times larger, then the area of ABC would be NINE times larger than DEF.

24. What is the slope of a line perpendicular to the line whose equation is 2y = −6x + 8?

1) -3
2) 1/6
3) 1/3
4) -6

Answer: 3) 1/3

Divide the entire equation by 2 to put it into slope-intercept form. That gives you y = -3x + 4.

The slope of the perpendicular line would have a slope that is the inverse reciprocal of -3, which is +1/3. That is Choice (3).

Another line with a slope of -3 would be parallel to this line (or it would be the same line).

Choosing either 1/6 or -6 means you aren't paying attention, didn't know what you were doing, or just guessing poorly.

More to come. Comments and questions welcome.

More Regents problems.

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