Tuesday, October 12, 2021

Geometry Problems of the Day (Geometry Regents, August 2012)

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 2012

Part I: Each correct answer will receive 2 credits.

16. Scalene triangle ABC is similar to triangle DEF. Which statement is false?

1) AB : BC = DE : EF
2) AC : DF = BC : EF
3) ∠ACB ≅ ∠DFE
4) ∠ABC ≅ ∠EDF

Answer: 4) ∠ACB ≅ ∠DFE

When two triangles are either congruent or similar, the corresponding angles are listed in order. Some instructors are not always consistent with this. (And, sometimes, I am carelessly guilty of this.)

So in this example, A corresponds to D, B corresponds to E and C corresponds to F.

This would mean that ∠ABC corresponds to and is congruent to ∠DEF, not ∠EDF.

17. Which equation represents a line that is parallel to the line whose equation is y = 3/2 x - 3 and passes through the point (1,2)?

1) y = 3/2 x + 1/2
2) y = 2/3 x + 4/3
3) y = 3/2 x - 2
4) y = -2/3 x + 8/3

Answer: 1) y = 3/2 x + 1/2

Parallel lines have the same slope. Eliminate Choices (2) and (4).

Substitute x = 1 into Choices (1) and (3) and see which one results in y = 2.

y = 3/2 (1) + 1/2 = 3/2 + 1/2 = 4/2 = 2. Choice (1) is the solution.

If you didn't have choices, you would have to solve for b, the y-intercept.

2 = 3/2 (1) + b
2 = 3/2 + b
b = 2 - 3/2 = 1/2

18. Lines a and b intersect at point P. Line c passes through P and is perpendicular to the plane containing lines a and b. Which statement must be true?

1) Lines a, b, and c are coplanar.
2) Line a is perpendicular to line b.
3) Line c is perpendicular to both line a and line b.
4) Line c is perpendicular to line a or line b, but not both.

Answer: 3) Line c is perpendicular to both line a and line b.

If line c is perpendicular to the plane, then it is perpendicular to all the lines contained within the plane that it intersects (at point P).

Choice (1) is not true because c is perpendicular to the plane containing a and b, so it can't be coplanar to them.

Choice (2) is not necessarily true, although it could be. We have no information about the relationship between lines a and b.

Choice (4) is not true. Line c cannot be perpendicular to only one of the two lines if both lines are contained in the plane that it is perpendicular to.

19. As shown in the diagram of triangle ACD below, B is a point on AC and DB is drawn.

If m∠A = 66, m∠CDB = 18, and m∠C = 24, what is the longest side of triangle ABD?

1) AB
2) DC
3) AD
4) BD

Answer: 1) AB

DC is NOT a side in triangle ABD, so eliminate Choice (2).

The longest side is opposite the biggest angle in the triangle. Angle A is 66 degrees, which is pretty big, but one of the other angles could be larger, so we need to calculate them.

Angle A = 66 and Angle C = 24, which is a total of 90 degrees. So Angle ADC must also be 90 degrees. Since Angle measures 18 degrees, then Angle ADB = 90 - 18 = 72. This is bigger than Angle A.

Common sense will tell you that ADB is the biggest angle because there is no way that ABD could be bigger than 72. It would be too much:

72 + 66 = 138, 180 - 138 = 42

Since ADB is the biggest angle, AB is the longest side.

20. In triangle ABC shown below, P is the centroid and BF = 18.

What is the length of BP?

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

Answer: 4) 12

The centroid is the point of concurrence (intersection of multiple lines) of the medians of a triangle. The centroid divides the median into two segments that have a ratio of 2:1.

If BF = 18, then BP is (2/3)(18) = 12.

Choice (1) 6 is the length of PF, which is 1/3 of 18.

Choice (2) 9 is half of BP, but the centroid doesn't divide the median in half.

Choice (3) 3 was probably included because the other choices were 6, 9 and 12.

More to come. Comments and questions welcome.

More Regents problems.

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