Thursday, December 02, 2021

Geometry Problems of the Day (Geometry Regents, August 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, August 2011

Part I: Each correct answer will receive 2 credits.

16. In the diagram below, AB is perpendicular to plane AEFG.

Which plane must be perpendicular to plane AEFG?


Answer: 1) ABCE

If AB is perpendicular to plane AEFG, then any plane containing AB must be perpendicular to plane AEFG.

Of the four choices, only ABCE contains AB.

There is nothing in the problem to state that this is a rectangular prism or that any of the planes are parallel.

17. How many points are both 4 units from the origin and also 2 units from the line y = 4?

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

Answer: 2) 2

This is a locus of points question. The locus of points 4 units away from the origin is a circle with radius 4 around the origin. The locus points two units away from y = 4 would be two horizontal lines, specifically y = 2 and y = 6.

The line y = 2 will intersect the circle in two places. The line y = 6 will not intersect the circle at all because the highest point on the circle will be (0,4), two units away from y = 6.

So the answer is 2.

18. When solved graphically, what is the solution to the following system of equations?

y = x2 − 4x + 6
y = x + 2

1) (1,4)
2) (4,6)
3) (1,3) and (4,6)
4) (3,1) and (6,4)

Answer: 3) (1,3) and (4,6)

You can put the equations into a graphing calculator and look at the graph or the table of values.

However, when you have a simple equation like y = x + 2, it's very easy to start eliminating bad choices. For example, 1 + 2 != 4, and 3 + 2 != 1, so Choices (1) and (4) can be eliminated.

Since (4,6) is in the two reason Choices, you only have to check (1,3) to see if the correct answer is (2) or (3).

(1)2 - 4(1) + 6 = 1 - 4 + 6 = 3, so (1,3) is a solution. Therefore, it is Choice (3).

Algebraically, you could solve the system of equations: equations?

y = x2 − 4x+ 6
y = x + 2

x2 - 4x + 6 = x + 2

x2 - 5x + 4 = 0

(x - 1)(x - 4) = 0

x - 1 = 0 or x - 4 = 0

x = 1 or x = 4

That would be enough to tell you that (3) is the correct answer, even without finding the matching y values.

15. Triangle PQR has angles in the ratio of 2:3:5. Which type of triangle is △PQR?

1) acute
2) isosceles
3) obtuse
4) right

Answer: 4) right

First of all, the triangle couldn't be isosceles. Besides the fact that the ratio 2:3:5 specifies that the triangle is scalene (with three different lengths), even if it could be isosceles, it would still have to be one of the other three choices.

You could solve the size of the angles with an equation like 2x + 3x + 5x = 180.

However, if you realize that the two smaller angles together equal the largest angle, then you should see that the largest angle must be 90 and the other two are complementary angles adding up to 90. So it is a right triangle.

You are given two angles and the reflexive property tells us that side AC ≅ AC. AC is NOT between the two pairs of angles, so it is AAS not ASA.

20. Plane A is parallel to plane B. Plane C intersects plane A in line m and intersects plane B in line n. Lines m and n are

1) intersecting
2) parallel
3) perpendicular
4) skew

Answer: 2) parallel

A plane intesects two other parallel planes at two parallel lines. Consider the bottom step and the top step in a flight of stairs between two parallel floors in a building.

If the planes are parallel, then lines contained within those planes could not intersect. Note that if they were perpenduclar, then they would also intersect, so (3) is automatically a bad choice.

Two lines in parallel planes could be skew but skew lines could not be connected by a single plane. This is the definition of skew.

More to come. Comments and questions welcome.

More Regents problems.

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