Geometry Problems of the Day (Geometry Regents, January 2012)

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 2012

Part II: Each correct answer will receive 2 credits. Partial credit is available.


29. In the diagram below, two parallel lines intersect circle O at points A, B, C, and D, with mAB = X + 20 and mDC = 2x - 20. Find m . A

Answer:

If two chords are parallel, then the arcs that they intercept will be congruent. Write an equation that says this, and solve it.

2x - 20 = x + 20
x = 40

AB = x + 20 = 40 + 20 = 60 degrees.

30. In the diagram below, point M is located on AB. Sketch the locus of points that are 1 unit from AB and the locus of points 2 units from point M.
Label with an X all points that satisfy both conditions.

Answer:

They are asking for a Sketch not a construction. Everything can be approximate, within reason.

The locus of points 1 unit away from AB is two parallel lines, one above and one below AB. AB should be right in the middle of them. It doesn't have to be exact, but don't be careless or sloppy about it.

The locus of points 2 units from M is a circle with M as its center. How big is "2 units"? Twice as big as the "1 unit" you already used. The important thing to realize is that the circle will overlap (and intersect) the parallel lines.

Your sketch should look something like this:

31. Determine whether the two lines represented by the equations y = 2x + 3 and 2y + x = 6 are parallel, perpendicular, or neither.
Justify your response.

Answer:

Rewrite the second equation to find its slope, or use the formula slope = -A/B.

The slope of the first line is 2. The slope of the second line is -1/2.

Since the two slopes are inverse reciprocals, with a product of -1, so the lines are perpendicular.

More to come. Comments and questions welcome.

More Regents problems.

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