Geometry Problems of the Day (Geometry Regents, August 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, August 2013

Part I: Each correct answer will receive 2 credits.


6. In triangle ABC, ∠A ≅ ∠B and ∠C is an obtuse angle. Which statement is true?

1) AC ≅ AB and BC is the longest side.
2) AC ≅ BC and AB is the longest side.
3) AC ≅ AB and BC is the shortest side.
4) AC ≅ BC and AB is the shortest side.

Answer: 2) AC ≅ BC and AB is the longest side.

If ∠A ≅ ∠B then triangle ABC is an isosceles triangle with AC ≅ BC. If ∠C is obtuse, then it is the largest angle of the triangle (because you can only have one obtuse angle in a triangle).

The side of the triangle opposite the largest angle is the longest side. The side opposite ∠C is AB.

7. In the diagram of ABC below, medians AD and BE intersect at point F.

If AF = 6, what is the length of FD?

1) 6
2) 2
3) 3
4) 9

Answer: 3) 3

Medians meet at the centroid. The centroid splits the median into two segments with lengths in the ratio of 2:1, that is, the longer segment from the vertex is twice as big as the shorter segment to the midpoint.

If AF = 6, the FD is half of that, or 3.

8. In circle O, diameter AB intersects chord CD at E. If CE = ED, then ∠CEA is which type of angle?

1) straight
2) obtuse
3) acture
4) right

Answer: 4) right

If a diameter bisects a chord, then the diameter is perpendicular to the chord. Therefore, ∠CEA is a right angle.

9. If triangle ABC ≅ triangle JKL ≅ triangle RST, then BC must be congruent to
1) JL
2) JK
3) ST
4) RS

Answer: 3) ST

Some instructors might be sloppy with this but when naming congruent triangles, the order of the letters matters.

In this example, ∠A ≅ ∠J ≅ ∠R, and AB ≅ JK ≅ RS

Therefore, BC ≅ KL ≅ ST

10. In the diagram of ABC below, AB is extended to point D.

If m∠CAB = x + 40, m∠ACB = 3x + 10, and m∠CBD = 6x, what is m∠CAB?

1) 13
2) 25
3) 53
4) 65

Answer: 4) 65

The Remote Angle Theorem tells us that the size of the exterior angle is equal to the sum of the two remote angles. You can write an equation to solve:

x + 40 + 3x + 10 = 6x
4x + 50 = 6x
50 = 2x
25 = x

If x = 25, then m∠CAB = x + 40 = 25 + 40 = 65.

More to come. Comments and questions welcome.

More Regents problems.

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