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.
1.Given: ABD, BC is the perpendicular bisector of AD
Which statement can not always be proven?
1) AC ≅ DC
2) BC ≅ CD
3) ∠ACB ≅ ∠DCB
4) Triangle ACB ≅ Triangle DCB
Answer: 2) BC ≅ CD
The keywords are "Perpendicular" and "bisector".
Choice (1) is true because AD is bisected. Eliminate this.
Choice (2) can not be proven. There is no correlation between the bisector and the segement being bisected.
Choice (3) is true because of BC is perpendicular to AD, and therefore ∠ACB and ∠DCB are both right angles. All right angles are congruent.
Choice (4) is true because of SAS. AC ≅ DC (see above), BC %cong; BC becasue of the Reflexive Property and ∠ACB ≅ ∠DCB )see above). Eliminate this.
2. In the diagram of circle O shown below, chord CD is parallel to diameter AOB and the measure of arc CD = 110.
What is the measure of arc DB?
1) 35
2) 55
3) 70
4) 110
Answer: 1) 35
If two chords are parallel then the arcs they intercept are congruent. Since AOB is a diameter then arc AB is a semicircle of 180 degrees.
Therefore, AC + CD + DB = 180
So x + 110 + x = 180
and 2x + 110 = 180
2x = 70
x = 35
The measure of arc DB is 35.
3. Given the statement:
What is the negation and the truth value of the negation?
1) One is not a prime number; true
2) One is not a prime number; false
3) One is a composite number; true
4) One is a composite number; false
Answer: 1) One is not a prime number; true
The negation of statement A is not A. It is not Z.
What I mean by this is that the negation of "prime" is "not prime". It is not "composite". There are other possibilities.
Likewise, the negation of "Even" is not "Odd". It's "not Even". Depending upon the types of numbers you are working with, a number that isn't Even might not be Odd either. It could be a mixed number, for example.
Also, one is NOT a prime number because it only has one factor. Prime numbers have exactly two factors.
4. Triangle ABC has the coordinates A(1,2), B(5,2), and C(5,5).
Triangle ABC is rotated 180° about the origin to form triangle A'B'C'.
Triangle A'B'C' is
1) acute
2) isosceles
3) obtuse
4) right
Answer: 4) right
The fact that the figure is rotated means nothing. Rotating a triangle doesn't change its shape.
The other important piece of information here is that any isosceles triangle MUST ALSO be acute, obtuse or right. Therefore, Choice (2) cannot be correct because one of the other choices would have to be correct as well.
You can plot the points on the graph paper and look. Odds are if you graph it accurately on actual graph paper, you could likely trust your eyes.
Mathematically, the easiest thing to check is if its a right triangle, so start there.
Points A and B both have a y-coordinate of 2. Therefore, AB is a vertical line. Points B and C both have an x-coordinate of 5. Therefore, BC is a horizontal line. AB is perpendicular to BC, which means that angle B is a right angle. So it's a right triangle.
5. What is an equation of the circle with center (-5,4) and a radius of 7?
1) (x - 5)2 + (y + 4)2 = 14
2) (x - 5)2 + (y + 4)2 = 49
3) (x + 5)2 + (y - 4)2 = 14
4) (x + 5)2 + (y - 4)2 = 49
Answer: 4) (x + 5)2 + (y - 4)2 = 49
Once again, the formula for the equation of a circle is (x - h)2 + (y - k)2 = r2, where (h, k) is the center of the circle, and r is the radius. Note that there are MINUS SIGNS in the formula.
The radius is 7, and 72 is 49, not 14. Eliminate Choices (1) and (3).
Since there is a minus sign in the formula, subtracting -5 is the same as ADDING 5.
More to come. Comments and questions welcome.
More Regents problems.
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