Geometry Problems of the Day (Geometry Regents, August 2013)

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 2013

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


32.In triangle ABC, the measure of angle A is fifteen less than twice the measure of angle B. The measure of angle C equals the sum of the measures of angle A and angle B. Determine the measure of angle B.

Answer:

The sum of the three angles is 180 degrees. Write an equation and solve it.

A = 2B + 15
C = A + B
A + B + C = 180
A + B + A + B = 180
2(A + B) = 180
2(2B + 15 + B) = 180
2(3B + 15) = 180
3B + 15 = 90
3B = 75
B = 25

This would make A equal to 2(25) + 15, or 65, and C = 25 + 65 = 90.

30. A circle has the equation (x - 3)² + (y + 4)² = 10. Find the coordinates of the center of the circle and the length of the circle’s radius.

Answer:

The center of the circle is (3, -4) and the radius is SQRT(10).

There isn't any work to show, and each answer is worth a point. It's important to remember to flip the signs in the equation to get the coordinates of the center.

34. Two intersecting lines are shown in the diagram below. Sketch the locus of points that are equidistant from the two lines. Sketch the locus of points that are a given distance, d, from the point of intersection of the given lines. State the number of points that satisfy both conditions.\

Answer:

A construction question and a locus of points question in the same Part II? (See yesterday's post.) Wow!

The points equidistant from these two lines would be two more lines going through the center of the X. Sketch a vertical and a horizontal line through the central point.

The locus of points of a distance, d, is a circle with a radius of d centered on the point of intersection of the lines. Draw a circle. It doesn't matter how big because it's only a sketch and you aren't told the size of d.

State the number of points that satisfiy both conditions: 4, the points on the circle that intersect the vertical and horizontal lines. (NOT the points on the diagonal lines.)

More to come. Comments and questions welcome.

More Regents problems.

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