Geometry Problems of the Day (Geometry Regents, June 2014)

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 2014

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


32. Two prisms with equal altitudes have equal volumes. The base of one prism is a square with a side length of 5 inches. The base of the second prism is a rectangle with a side length of 10 inches.

Determine and state, in inches, the measure of the width of the rectangle.

Answer:

The area of the square must equal the area of the rectangle.

5 * 5 = 10 * w
25 = 10w
w = 2.5

Yes, it's that simple. One credit for the work and one for the solution. You probably could have done the work in your head, but put it on paper anyway.

33.As shown in the diagram below, BO and tangents BA and BC are drawn from external point B to circle O. Radii OA and OC are drawn.

If OA = 7 and DB = 18, determine and state the length of AB.

Answer:

First thing to remember is that every radius is the same size, so if you know one of them, you know all of them. Write them down on the paper so you have it in front of you.

Second thing is that tangents to circles make right angles with radii, so you have right triangles in this figure, and you can use Pythagorean Theorem.

OA = 7, OD = 7, DB = 18, so AB = 25.

7² + (AB)² = 25²
49 + x² = 625
AB² = 576
AB = 24

7, 24, 25 is a Pythagorean triple you should be familiar with if only to speed up problems like this one.

34. Triangle RST is similar to XYZ with RS = 3 inches and XY = 2 inches. If the area of triangle RST is 27 square inches, determine and state the area of trianlge XYZ, in square inches.

Answer:

If RST ~ XYZ then side RS corresponds to side side XY. The scale factor between the two triangles is 3/2.

This means that that base of XYZ is 2/3 of the base of RST and the altitude of XYZ will be 2/3 of the altitude of RST. So the Area of XYZ = (2/3)(2/3)Area of RST.

Area XYZ = (4/9)(27) = 12 square inches.

End of Part II.

More to come. Comments and questions welcome.

More Regents problems.