Geometry Problems of the Day (Geometry Regents, January 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, January 2014

Part I: Each correct answer will receive 2 credits.


16. For which measures of the sides of tiangle ABC is angle B the largest angle of the triangle?

1) AB = 2, BC = 6, AC = 7
2) AB = 6, BC = 12, AC = 8
3) AB = 16, BC = 9, AC = 10
4) AB = 18, BC = 14, AC = 5

Answer: 1) AB = 2, BC = 6, AC = 7

The longest side in a triangle is opposite the largest angle, so for B to be the largest angle, it has to be across from the largest side. The side opposite angle B is side AC.

In Choice (1), AC is the longest side, so B is the largest angle.

In Choice (2), BC is the longest side, so A is the largest angle.

In Choice (3), AB is the longest side, so C is the largest angle.

In Choice (4), AB is the longest side, so C is the largest angle.

17. What is the measure of the largest exterior angle that any regular polygon can have?

1) 60°
2) 90°
3) 120°
4) 360°

Answer: 3) 120°

The sum of the exterior angles of a polygon is ALWAYS 360°. The more sides that a polygon has, the more exterior angles it has. The more exterior angles a regular polygon has, the smaller each individual exterior angle will be.

What you need is the polygon with the fewest sides, and that is a triangle. A regular triangle (an equilateral triangle) has three 60° interior angles. Therefore, the exterior angles will be 120° each.

18. As shown in the diagram below, a landscaper uses a cylindrical lawn roller on a lawn. The roller has a radius of 9 inches and a width of 42 inches

To the nearest square inch, the area the roller covers in one complete rotation is

1) 2,374
2) 2,375
3) 10,682
4) 10,688

Answer: 2) 2,375

They are asking for the lateral area of a cylinder. That is, if you imagine a soup can, they want the area of the rectangular label wrapped around it, but not that top or bottom of the can.

The area of a rectangle is length times width. The width is 42 inches. The length if the circumference of a circle with radius 9, with is 2 * pi * (9).

SA = 42 * 18 * 3.141592 = 2375.04...

This is a slighty EVIL question because if you used 3.14 for pi, the rounding error would give you 2,374 as an answer because you would not have used enough precision. Thankfully, your calculator should have a "PI" key.

19. In the diagram below, AC and BC are tangent to circle O at A and B, respectively, from external point C.

If m∠ACB = 38, what is m∠AOB?

1) 71
2) 104
3) 142
4) 161

Answer: 3) 142

AOBC is a quadrilateral and its interior angles sum up to 360 degrees. Angles A and B are right angles, 90 degrees each, because they are tangent to the circle.

So 38 + 90 + 90 + x = 360
218 + x = 360
x = 142

∠AOB is 142 degrees.

I don't see how they arrived at the incorrect choices. I don't see any obvious mistakes that might cause them, except that 71 is half of 142.

15. What is the perimeter of a square whose diagonal is 3 SQRT(2)?

1) 18
2) 12
3) 9
4) 6

Answer: 2) 12

Think about Pythagorean Theorem.

(side)² + (side)² = (diagonal)²

So 2x² = (3 * SQRT(2))²
2x² = 9 * 2
2x² = 18
x² = 9
x = 3

Since one side = 3, then the perimeter = 12.

More to come. Comments and questions welcome.

More Regents problems.