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 I: Each correct answer will receive 2 credits.


21. The image of rhombus VWXY preserves which properties under the transformation T_2,-3?

1) parallelism, only
2) orientation, only
3) both parallelism and orientation
4) neither parallelism nor orientation

Answer: 3) both parallelism and orientation

A translation is a Rigid Transformation, meaning that the size, shape and orientation of the object is preserved in its image. The sides will still be parallel and the orientation will not change.

22. The equation of a circle is (x - 3)² + y² = 8. The coordinates of its center and the length of its radius are

1) (-3, 0) and 4
2) (3, 0) and 4
3) (-3, 0) and 2 SQRT(2)
4) (3, 0) and 2 SQRT(2)

Answer: 4) (3, 0) and 2 SQRT(2)

The general form is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Since there is no k, it is 0. The square root of 8 is NOT 4, so eliminate Choices (1) and (2).

There is a minus sign in the form, so h is the number after the sign, which is 3.

23. Which statement has the same truth value as the statement “If a quadrilateral is a square, then it is a rectangle”?

1) If a quadrilateral is a rectangle, then it is a square.
2) If a quadrilateral is a rectangle, then it is not a square.
3) If a quadrilateral is not a square, then it is not a rectangle.
4) If a quadrilateral is not a rectangle, then it is not a square.

Answer: 4) If a quadrilateral is not a rectangle, then it is not a square.

The Contrapositive has the same truth as the original conditional statement.

To get the contrapositive, you negate both parts of the conditional and then reverse the order. In other words, you take the inverse of the converse (or vice versa).

Choice (1) is the converse. Eliminate it.

Choice (2) is neither the converse or the contrapositive. It is an unrelated statement. Eliminate it.

Choice (3) is the inverse. Eliminate it.

Choice (4) is the contrapositive, which is the correct answer.

24.The three medians of a triangle intersect at a point. Which measurements could represent the segments of one of the medians?

1) 2 and 3
2) 3 and 4.5
3) 3 and 6
4) 3 and 9

Answer: 3) 3 and 6

Three medians of a triangle intersect at the centroid. The centroid will divide each median into two segments, one of which will be twice the length of the other. (The segment from the vertex to the centroid will always be the longer side, 2/3 of the length of the median.)

Only choice (3) has one segment that is twice the size of the other.

20. In the diagram of PQR shown below, PR is extended to S, m∠P = 110, m∠Q = 4x, and m∠QRS = x² + 5x.

What is m∠Q?

1) 44
2) 40
3) 11
4) 10

Answer: 2) 40

The exterior angle is the sum of the two remote angles. Write an equation that states this. Then solve the quadratic by setting it equal to 0 and factoring. Throw out the negative result.

x² + 5x = 4x + 110
x² + 5x - 4x - 110 = 0
x² + x - 110 = 0
(x + 11)(x - 10) = 0
x + 11 = 0 or x - 10 = 0
x = -11 or x = 10

The value of angle Q is 4x (not just x). Since 4(-11) or -44 degrees is not a valid amount for an angle in a triangle, discard that result.

Therefore, m∠Q = 4(10) = 40.

Note that the other choices are 10, 11, and 4(11), so it's easy to pick the incorrect choice.

More to come. Comments and questions welcome.

More Regents problems.