Geometry Problems of the Day (Geometry Regents, June 2013)

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 2013

Part I: Each correct answer will receive 2 credits.


21. In triangle ABC, m∠A = 60, m∠B = 80, and m∠C = 40. Which inequality is true?
AB > BC
2) AC > BC
3) AC < BA
4) BC < BA

Answer: 2) AC > BC

It may not look obvious, but Choices (1) and (4) are the same thing. Either they are both true or they are both false, but they cannot both be true.

For any triangle the longest side is opposite the biggest angle, and the smallest side is opposite the smallest angle. If two angles are congruent, then the opposite sides are congruent.

If ∠B is the largest angle then side AC is the longest side, which means it's bigger than BC, so Choice (2) is correct.

22. Circle O with ∠AOC and ∠ABC is shown in the diagram below.

What is the ratio of m∠AOC to m∠ABC?

1) 1:1
2) 2:1
3) 3:1
4) 1:2

Answer: 2) 2:1

∠AOC is a central angle and ∠ABC is an inscribed angle. The relationship between the two is that a central angle is twice the size of an inscribed angle that intercepts the same arc (in this case, arc AC).

So the ratio of ∠AOC to ∠ABC is 2:1. Note that the order in the question is important. It is NOT 1:2. That would be the ratio of ∠ABC to ∠AOC.

23. A rectangular prism has a base with a length of 25, a width of 9, and a height of 12. A second prism has a square base with a side of 15. If the volumes of the two prisms are equal, what is the height of the second prism?

1) 6
2) 8
3) 12
4) 15

Answer: 3) 12

Calculate the Volume of the first prism, and then solve for the height of the second one.

V = (25)(9)(12) = 2700

V = (15)(15)H = 2700
H = 2700 / 225 = 12

The slopes are not the same and they are not inverse reciprocials (that is, they do not have a product of -1). Therefore, they intersect but are not perpendicular.

24. In triangles ABC and DEF, AB = 4, AC = 5, DE = 8, DF = 10, and ∠A ≅ ∠D. Which method could be used to prove triangle ABC ∼ triangle DEF?

1) AA
2) SAS
3) SSS
4) ASA angle.

Answer: 2) SAS

When you see similarity, your first thought is to go to AA, but you are only given one angle here.

However, you are given two pairs of sides and the included angle, which is SAS. The corresponding sides are proportional. so the triangles are similar.

More to come. Comments and questions welcome.

More Regents problems.

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