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 2014
Part I: Each correct answer will receive 2 credits.
26. Triangle PQT with RS || QT is shown below.
If PR = 12, RQ = 8, and PS = 21, what is the length of PT?
1) 14
2) 17
3) 35
4) 38
Answer: 3) 35
Since RS || QT, then triangles PRS and PQT are similar. That means that their corresponding angles are congruent and their corresponding sides are proportional. (And the ratio of two corresponding side would be the scale factor.)
Remember that PR corresponds to PQ, but you weren't given PQ, you were given RQ. PQ = PR + RQ = 12 + 8 = 20.
Also note that they asked for PT, not ST.
Set up a proportion and cross-multiply:
12 / 20 = 21 / PT
12 PT = 420
PT = 35
Note that Choice (1) 14 is the length of ST.
27. In the diagram of WX computations. YZ below, WY ≅ XZ.
Which reasons can be used to prove Wx ≅ YZ?
1) reflexive property and addition postulate
2) reflexive property and subtraction postulate
3) transitive property and addition postulate
4) transitive property and subtraction postulate
Answer: 2) reflexive property and subtraction postulate
The reflexive property says that XY ≅ XY. If you remove XY from each of the two congruent lines, what remains will be congruent because of the subtraction postulate.
Consider this: You and a friend have the same amount of money. It doesn't matter how much that is. You both buy the same ticket to the same movie at the same theater. You each buy the same size popcorn and the same drink. Afterward, you will both still have the same amount of money although it will be considerably less than it was before.
28. The coordinates of the endpoints of the diameter of a circle are (2,0)
and (2,-8). What is the equation of the circle?
1) (x - 2)2 + (y + 4)2 = 16
2) (x + 2)2 + (y - 4)2 = 16
3) (x - 2)2 + (y + 4)2 = 8
4) (x + 2)2 + (y - 4)2 = 8
Answer: 1) (x - 2)2 + (y + 4)2 = 16
Another equation of a circle question. There were a few in Part I of this test. Must be important!
If the two endpoints are (2, 0) and (2, -8) then the circle of the circle must be in the middle, which is (2, -4). It was nice of them to make it a simple vertical line. If you couldn't tell this, you could have made a simple sketch on the graph paper in the back of the booklet.
The general formula for the equation of a circle is (x - h)2 + (y - k)2 = r2, where (h, k) is the center of the circle.
Because -4 is already negative, -(-4) becomes + 4. Eliminate Choices (2) and (4).
The radius is 4 because 0 - (-4) = 4 and (-4) - (-8) = 4. Or count the boxes on your sketch. The diameter is 8 and half of 8 is 4. Square 4 and you get 16. That's Choice (1).
End of Part I.
More to come. Comments and questions welcome.
More Regents problems.
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