Geometry Problems of the Day (Geometry Regents, August 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, August 2011

Part I: Each correct answer will receive 2 credits.


21. The diagonals of a quadrilateral are congruent but do not bisect each other. This quadrilateral is

1) an isosceles trapezoid
2) a parallelogram
3) a rectangle
4) a rhombus

Answer: 1) an isosceles trapezoid

You can make sketches if you aren't sure. One of the characteristics of parallelograms is that the diagonals bisect each other, and Rectangles and Rhombuses are parallelograms, so they have this property as well.

The diagonals of isosceles trapezoids will not bisect each other.

Note: If you are a professor of mathematics who would like to argue that trapezoids have at least one pair of parallel sides, please argue with the Regents people, not me. I'll politely laugh in your general direction for wasting your time tilting at quadrilaterals. Especially if you use tau, too.

22. What is the slope of a line perpendicular to the line represented by the equation x + 2y = 3?

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

Answer: 2) 2

A line perpendicular to the given line will have a slope that is the inverse reciprocal of the slope of the given line. So first you need to know the slope of the given line.

The slope of a line in standard form, Ax + By = C, is -A/B, which in this case is -1/2. If you forgot that (or didn't know it), you can rewrite the equation in slope-intercept form:

x + 2y = 3
2y = -x + 3
y = -1/2 x + 3/2

The slope of the original line is -1/2, so the perpendicular line has a slope of 2.

23. WA packing carton in the shape of a triangular prism is shown in the diagram below.

What is the volume, in cubic inches, of this carton?

1) 20
2) 60
3) 120
4) 240

Answer: 3) 120

A triangular prism has a volume that is 1/2 the size of a rectangular prism of the same dimensions, just like with area are triangles and rectangles.

V = 1/2 L * W * H = 1/2 (4)(6)(10) = 120

24. In the diagram below of circle O, diameter AOB is perpendicular to chord CD at point E, OA = 6, and OE = 2.

What is the length of CE?

1) 4 √3
2) 2 √3
3) 8 √2
4) 4 √2

Answer: 4) 4 √2

You can use Pythagorean Theorem or the Intersecting Chords Theorem.

Draw radius OC, which has a length of 6. It is the hypotenuse of OEC. We then know that

2² + (CE)² = 6²

4 + (CE)² = 36

CE = √32 = √16 * √2 = 4 √2

Or you know that CE = DE and AE = AO + OE = 6 + 2 = 8 and BE = 6 - 2 = 4.

So (CE)² = (8)(4) = 32, which gives the same result as above.

More to come. Comments and questions welcome.

More Regents problems.

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