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.
6. When the system of equations y + 2x = x2 and y = x is graphed on
a set of axes, what is the total number of points of intersection?
1) 1
2) 2
3) 3
4) 0
Answer: 2) 2
A straight line and a parabola can have 0, 1, or 2 intersection points. They can NEVER have 3. Eliminate Choice (3).
Both equations have a y-intercept of 0, so there is at least one point of intesection at (0, 0). The question becomes, is there a second? You probably already "know" there is, but let's make certain.
If you graph y = x and y = x2 - 2x, you will see that they have 2 intersection points.
If you wanted to solve this algebraically, you can use Substitution:
x = x2 - 2x
0 = x2 - 3x
0 = x (x - 3)
The equations will intersect at when x = 0 and x = 3. Since y = x, the points of intersection will be (0,0) and (3, 3).
7. The vertex angle of an isosceles triangle measures 15 degrees more
than one of its base angles. How many degrees are there in a base angle
of the triangle?
1) 50
2) 55
3) 65
4) 70
Answer: 2) 55
You can right an equation or you can check each of the choices. Remember that there are TWO base angles and ONE vertex angle.
50 + 50 + 65 = 165, not a triangle. Eliminate Choice (1).
55 + 55 + 70 = 180. This is a triangle. Choice (2) is the answer.
65 + 65 + 80 = 210, not a triangle. Eliminate Choice (3).
70 + 70 + 85 = 225, not a triangle. Eliminate Choice (4).
Or you can set up an equation:
3x + 15 = 180
3x = 155
x = 55
There are 55 degrees in the Base Angle, which means that there are 70 degrees in the vertex. Make sure you select the correct answer.
8. Circle O is graphed on the set of axes below. Which equation
represents circle O?
1) (x + 1)2 + (y - 3)2 = 9
2) (x - 1)2 + (y + 3)2 = 9
3) (x + 1)2 + (y - 3)2 = 6
4) (x - 1)2 + (y + 3)2 = 6
Answer: 1) (x + 1)2 + (y - 3)2 = 9
The formula for a circle is (x - h)2 + (y - k)2 = r2, where (h, k) is the center of the circle.
Note that the radius is 3 and that 32 = 9. Eliminate Choice (3) and (4).
The point (h, k) is (-1, 3), so the correct choice is (1).
Choice (2) does not have the signs reversed.
9. In the diagram of the circle shown below, chords AC and BD intersect at Q, and chords AE and BD are parallel.
Which statement must always be true.
1) Arc AB ≅ Arc CD
2) Arc DE ≅ Arc CD
3) Arc AB ≅ Arc DE
4) Arc BD ≅ Arc AE
Answer: 3) Arc AB ≅ Arc DE
If two chords in a circle are parallel, then the arcs that they intercept will be congruent. This can be shown by drawing, for example, chord AD, which would be a transversal across BD and AE. The alternate interior angles would be congruent and therefore the arcs that they intercept would be congruent.
There is no guarantee that arcs AB and CD will be the same size. It would only happen if QB ≅ QC and QA ≅ QD.
10. In the diagram below, triangle AEC ≅ triangle BED.
Which statement is not always true?
1) AC ≅ BD
2) CE ≅ DE
3) ∠ EAC ≅ ∠ EBD
4) ∠ ACE ≅ ∠ DBE
Answer: 4) ∠ ACE ≅ ∠ DBE
This question relies on the fact that you understand that triangle AEC ≅ triangle BED means that ∠A ≅ ∠B, ∠E ≅ ∠E, and ∠C ≅ ∠D. Otherwise, you are going on what LOOKS to be correct, which you should never rely on.
AC and BD are corresponding sides of congruent triangles, so they are always congruent.
CE and DE are corresponding sides of congruent triangles, so they are always congruent.
∠EAC and ∠EBD are corresponding angles of congruent triangles, so they are always congruent.
∠ACE and ∠DBE are NOT corresponding angles of congruent triangles. They do NOT have to be congruent. This is the answer.
More to come. Comments and questions welcome.
More Regents problems.
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