Tuesday, October 19, 2021

Geometry Problems of the Day (Geometry Regents, June 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, June 2012

1. Triangle ABC is graphed on the set of axes below.

Which transformation produces an image that is similar to, but not congruent to, triangle ABC?

1) T2,3
2) D2
3) ry = x
4) R90

A Dilation does not preserve distance, or size, by defintion. The shape will be conserved, meaning that all of the angles in the image are congruent to the original, which is the definition of similar.

Translations, reflections and rotations are produce images that are congruent to the original pre-image.

2. A student wrote the sentence “4 is an odd integer.” What is the negation of this sentence and the truth value of the negation?

1) 3 is an odd integer; true
2) 4 is not an odd integer; true
3) 4 is not an even integer; false
4) 4 is an even integer; false

Answer: 2) 4 is not an odd integer; true

The negation of "is" is "is not", and the rest of the sentence doesn't matter.

The negation of this sentence is "4 is not an odd integer" and that negation is true.

Negative does not make 4 into 3. Why 3? Why not any other number? This makes no sense. A negation is very specific.

Changing "is an odd" to "is not an even" requires changing TWO things: adding the "not" and switching "odd" to "even". This is not how a negation is done.

With Choice (4), it could be argued argued that "not an odd integer" is the same as "an even integer". However, this ignores the case that it might not be an integer at all, which would mean that the number is neither odd nor even. There are more than two possibilities. In any event, the statment "4 is an even integer" is true, not false, so the Choice is incorrect regardless of any argument.

3. As shown in the diagram below, EF intersects planes P , Q , and R/

If EF is perpendicular to planes P and R , which statement must be true?

1) Plane P is perpendicular to plane Q.
2) Plane R is perpendicular to plane P.
3) Plane P is parallel to plane Q.
4) Plane R is parallel to plane P.

Answer: 4) Plane R is parallel to plane P.

If you picture EF as a horizontal line, it would be perpendicular to vertical planes. That would make planes P and R parallel to each other.

Plane Q has no special relationship with either planes P or R, nor with line EF. Plane Q is shown to intersect P and R, so it cannot be parallel to them. If it were perpendicular to P and R, then plane Q would either have to be parallel to EF, which is can't because they intersect, or it would have to contain the line EF in its entirety. It isn't shown to do that, and the word "intersect" would suggest that this isn't the case.

4. In the diagram below, LATE is an isosceles trapezoid with LE ≅ AT, LA = 24, ET = 40, and AT = 10. Altitudes LF and AG are drawn.

What is the length of LF?

1) 6
2) 8
3) 3
4) 4

Since it is an isosceles trapezoid, you know that triangles LEF and AGT are congruent (by AAS, if you need to work it out). That means that EF ≅ GT. Call each of those x.

Also, FG ≅ LA.

So x + 24 + x = 40, meaing 2x + 24 = 40. Then x = 8.

So EF = 8, LE = 10 and LFE is a right triangle.

A quick use of Pythagorean Theorem tells you that LF = 6. And you should seriously know 6-8-10 is a Pythagorean Triple without needing to use the theorem.

remain parallel.

5. In the diagram below of circle O, diameter AB is parallel to chord CD.

If mCD = 70, what is mAC?

1) 110
2) 70
3) 55
4) 35

AC = BD and AC + CD + BD = 180

So 2AC + 70 = 180
2AC = 110
AC = 55

More to come. Comments and questions welcome.

More Regents problems.

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