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


29. In the diagram below of △ACD, B is a point on AC such that △ADB is an equilateral triangle, and △DBC is an isosceles triangle with DB ≅ BC. Find m∠C.

Answer:

Look at the image below:

All of the angles in the equilateral triangle are 60 degrees. Angles ABD and DBC are supplementary, so angle DBC is 120 degrees.

Angle DBC is also the vertex angle of the isosceles triangle making angles BDC and C the base angles, which must be congruent.

180 - 120 = 60 and 60 / 2 = 30, so m∠C = 30.

30. Triangle ABC has vertices A(−2,2), B(−1,−3), and C(4,0). Find the coordinates of the vertices of △A′B′C′, the image of △ABC after the transformation r_x-axis.
[The use of the grid below is optional.]

Answer:

A reflection across the x-axis would flip the sign of the y-coordinate. If the original is positive, then the image is negative and vice versa. A point on the x-axis will reflect onto itself. If this doesn't seem correct to you, then graph it.

A(−2,2) is reflected to A'(-2,-2).

B(−1,−3) is reflected to B'(-1,3).

C(4,0) is reflected onto itself, C'(4, 0).

31. Find, in degrees, the measures of both an interior angle and an exterior angle of a regular pentagon.

Answer:

A pentagon has five sides and angles. The total number of degrees in any pentagon is (5 - 2)*180 = 540.

In a regular pentagon, each angle is the same size, so 540 / 5 = 108 degrees for each interior angle.

The exterior angles are supplementary, so 180 - 108 = 72 degrees.

You could also do these in the opposite order.

The sum of the exterior angles of any convex polygon is 360 degrees. Therefore each individual exterior angle in a regular pentagon must be 360 / 5 = 72 degrees.

Each interior angle is supplementary to the exterior angle, so 180 - 72 = 108 degrees.

More to come. Comments and questions welcome.

More Regents problems.

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