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 I: Each correct answer will receive 2 credits.


1. In the diagram below, AB, BC, and AC are tangents to circle O at points F, E, and D, respectively, AF = 6, CD = 5, and BE = 4.

What is the perimeter of △ABC?

1) 15
2) 25
3) 30
4) 60

Answer: 3) 30

When two tangents of a circle intersect, the segments between the point of intersection and the circle will be congruent. So AD ≅ AF; BE ≅ BF; CD ≅ CE.

So the perimeter is 6 + 6 + 5 + 5 + 4 + 4 = 30, or 2 * (6 + 5 + 4) = 30.

2. Quadrilateral MNOP is a trapezoid with MN || OP. If M′N′O′P′ is the image of MNOP after a reflection over the x-axis, which two sides of quadrilateral M′N′O′P′ are parallel?

1) M'N' and O'P'
2) M'N' and N'O'
3) P'M' and O'P'
4) P'M' and N'O'

Answer: 1) M'N' and O'P'

A reflection doesn't change the shape of the object. Lines that were parallel will still be parallel in the image.

If MN || OP in the pre-image, then M'N' || O'P' in the image.

3. In the diagram below of △ABC, D is the midpoint of AB, and E is the midpoint of BC.

If AC = 4x + 10, which expression represents DE?

1) x + 2.5
2) 2x + 5
3) 2x + 10
4) 8x + 20

Answer: 2) 2x + 5

The midsegment is half of the length of the side of the triangle that it is parallel to.

Half of 4x + 10 is 2x + 5.

4. Which statement is true about every parallelogram?

1) All four sides are congruent.
2) The interior angles are all congruent.
3) Two pairs of opposite sides are congruent.
4) The diagonals are perpendicular to each other.

Answer: 3) Two pairs of opposite sides are congruent

The opposite sides of a parallelogram are always congruent.

All four sides will be congruent only in a rhombus (including squares). The interior angles are all congruent only in rectangles (including squares). The diagonals are perpendicular only in rhombuses.

5. The diagram below shows a rectangular prism.
Which pair of edges are segments of lines that are coplanar ?

1) AB and DH
2) AE and DC
3) BC and EH
4) CG and EF

Answer: 3) BC and EH

The edges of a prism that are coplanar will be pointed in the same direction. If they aren't pointed in the same direction, they will be skew.

AB goes front to back, but DH goes up and down. Eliminate Choice (1).

AE goes up and down, but DC goes front to back. Eliminate Choice (2).

BC goes left to right, and EH goes left to right. This is the correct answer.

CG goes up and down, EF goes front to back. Eliminate Choice (4).

at long before the end.

More to come. Comments and questions welcome.

More Regents problems.

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