A common theme: there were several questions involving the equation of a circle, as there always are, as well as several questions which not only involved the Pythagorean Theorem, but very specifically a 3-4-5 triangle or a multiple of it.
I hope everyone did well. Part I was, in my opinion, very "get-able" -- not very difficult for a Regents exam.
As always, apologies in advance for typos. I did rush this a bit. Here is Part I.
1. What is the solution of the system of equations graphed below? (Image omitted)
y = 2x + 1
y = x2 + 2x – 3
The solutions of a system are the points where the two lines intersect. In this problem, that would be at (-2, -3) and (2, 5). Choice (4).
If you look at the other choices, (0, -3) is the y-intercept of the parabola, (-1, -4) is the vertex and (-3,0) and (1, 0) are the roots of the parabola. None of those have anything to do with the linear equation.
2. What are the coordinates of the midpoint of the line segment with endpoints (2, -5) and (8, 3)?
Take the average of the x-coordinates and of the y-coordinates. You get (5, -1) which is exactly in the middle. Alternatively, you could have used the scratch graph paper and plotted it instead of calculating.
3. As shown in the diagram below, when hexagon ABCDEF is reflected over line m, the image is hexagon A’B’C’D’E’F’. Under this transformation, which property is not preserved? (Image omitted)
Orientation is not preserved. The object has the same shape and size, but the direction its facing switches.
4. In the diagram of triangle ABC below, BD is drawn to side AC. (Image omitted)
If m<A = 35, m<ABD = 25, and m<C = 60, which type of triangle is BCD?
BCD is equilateral. Angles A and ABD add up to 60 degrees. This makes <ADB = 120 degrees. Angle BDC is supplementary to ADB, so it is 60 degrees. Angle C is 60 degrees as well, so <CBD must be 60 degrees. Triangle BCD has three 60-degree angles, so it is an equilateral triangle.
5. In the diagram below of rhombus ABCD, the diagonals AC and BD intersect at E. (Image omitted)
If AC = 18 and BD = 24, what is the length of one side of rhombus ABCD?
The diagonals of a rhombus are perpendicular and bisect each other. They form four congruent right triangles. Each side of the rhombus is the hypotenuse of one of those right triangles. AC = 18, so AE = 9. BD = 24, so BE = 12. If the two legs of a right triangle are 9 and 12, then the hypotenuse is 15. Note: You can use the Pythagorean Theorem, but you should know 9-12-15 because it’s a multiple of 3-4-5, which is the most common right triangle used in these problems! The answer is 15.
6. What are the truth values of the statement “Opposites angles of a trapezoid are always congruent.” and its negation?
The statement is false, which makes the negation (“Opposite angles of a trapezoid are not always congruent.” ) true.
7. What is the length of a line segment whose endpoints have coordinates (5, 3) and (1, 6)?
Use the distance formula, or sketch the line segment, make a right triangle and use Pythagorean Theorem. Either way, you will get the square root of (32 + 42), which is the square of (9 + 16), which is 5. Again, 3-4-5 is a common right triangle. Recognize it when you see it!
8. In the diagram below of isosceles triangle ABC, the measures of vertex angle B is 80 degrees. If AC extends to point D, what is m<BCD? (Image omitted)
The vertex angle is 80 degrees, so the two base angles have a total of 100 degrees. The base angles are congruent, so each is 50 degrees. Angle BCD is supplementary to a 50 degree angle, so it is 130 degrees.
9. A student used a compass and a straightedge to construct CE in triangle ABC as shown below. (Image omitted)
Which statement must always be true for this construction?
The construction shows the bisecting of angle C. So <ACE is congruent to < BCE.
10. In triangle, ABC, AB = 4, BC = 7, and AC = 10. Which statement is true?
The smallest angle is across from the smallest side and the biggest angle is across from the biggest side. (Triangle Inequality Theorem). The opposite angle will be the vertex that is NOT part of the line segment; i.e., <C is across from AB and is the smallest angle, and <B is the largest. The correct choice is (2) m<B > m <A > m< C. (Sorry about the formatting. It can’t be helped.)
11. A circle whose center has coordinates (-3, 4) passes through the origin. What is the equation of this circle?
Using Pythagorean Theorem, we can see that the radius of the circle is 5. Again, another 3-4-5 triangle!
Flip the signs and square the radius: (x + 3)2 + (y – 4)2 = 25
12. Point W is located in plane R. How many distinct lines passing through point W are perpendicular to plane R?
One. Think of any vertical pole coming out of the ground. There’s only one possible vertical pole at that point in the ground.
13. In the diagram below (image omitted), line l is parallel to line m, and line w is a transversal.
If m<2 = 3x + 17 and m <3 = 5x – 21, what is m<1?
Angles 2 and 3 are supplementary. Angles 1 and 2 are congruent. Solve for x and then find the measure of angle 2 to get the answer.
3x + 17 + 5x – 21 = 180
8x – 4 = 180
8x = 184
x = 23
m<2 = 3(23) + 17 = 69 + 17 = 86. Choice (4).
Note: be careful. I subtracted 4 instead of adding first time through. My incorrect answer wasn’t one of the choices, so I knew to go back and check my work!
14. The diagram below is of circle O. (Image omitted – it has a circle with center O at (5, -3) and a radius of 4.)
Which equation represents circle )?
Again. Flip the signs and square the radius. (x – 5)2 + (y + 3)2 = 16.
15. In isosceles trapezoid QRST show below, QR and TS are bases. (Image omitted)
If m<Q = 5x + 3 and m<R = 7x – 15, what is m<Q?
Angles Q and R are congruent, so their measures are equal.
5x + 3 = 7x – 15
3 = 2x – 15
18 = 2x
9 = x
m<Q = 5(9) + 3 = 45 + 3 = 48. Choice (2).
16. Triangle ABC is graphed on the set of axes below. (Image omitted)
What are the coordinates of the point of intersection of the medians of triangle ABC?
This one is easier than it sounds. The median from vertex A to the midpoint of BC is a horizontal line with a length of 6. The medians will meet at the centroid, which is 2/3 of the way from A to the midpoint. That means that it will be 4 units to the right of point A. The answer is (-1, 2).
If you drew the three medians (two would be enough, though), you could probably eyeball the correct answer from the choices given. Only one really makes sense.
17. Given the statement, “If a number has exactly two factors, it is a prime number,” what is the contrapositive of this statement?
Negative both parts and flip it around. “If a number is not a prime number, it does not have exactly two factors.” Choice (2).
18. Which graph represents a circle whose equation is (x – 2)2 + (y + 4)2 = 4. (Images omitted)
Third time! Flip the signs. The center is (2, -4). The radius is 2, which is the square root of 4. That makes Choice (3) the correct answer.
19. If two sides of a triangle have lengths of 4 and 10, the third side could be …
The third side must be greater than 6, which is 10-4, and less than 14, which is 10+4. The only choice that fits is Choice (1) 8.
20. The lines represented by the equations 4x + 6y = 6 and y = 2/3x – 1 are …
Put the first line into slope-intercept form:
4x + 6y = 6
6y = -4x + 6
y = -2/3x + 1
The lines have the different slope, so they are neither parallel nor the same line. Additionally, the slopes are not negative reciprocals, so they are not perpendicular. However, they will intersect.
21. In the diagram below of triangle ABC, DE || AB. (Image omitted)
If CD = 4, CA = 10, CE = x + 2 and EB = 4x – 7, what is the length of CE?
Set up a proportion. CD/DA = CE/EB. Notice that I said DA, not CA. DA = 10-4 = 6.
So 4/6 = x+2/4x -7. Cross multiply add you get
4(4x – 7) = 6(x + 2)
16x – 28 = 6x + 12
10x = 40
X = 4
CE = 4 + 2 = 6. Choice (3).
22. Parallelogram ABCD with diagonals AC and BD intersecting at E is shown below (image omitted).
Which statement must be true?
Diagonals of a parallelogram bisect each other, but that does not appear in the choices.
Each diagonal is a transversal across two parallel lines, so the alternate interior angles are congruent. This is needed to prove choice (2).
23. In the diagram below of circle O, m<ABC = 24. (Image omitted)
What is m<AOC?
The inscribed angle is half the size of the central angle. If m<ABC is 24, then m<AOC = 48.
24. Triangle A’B’C’ is the image of triangle ABC after a dilation of 2. Which statement is true?
The original triangle will have sides that are half as long as the image. The angles will be the same. So the angle is choice (3), m<B = m<B’.
25. In the diagram of the circle below, AD || BC, arc AB = (5x + 30) degrees, and arc CD = (9x – 10) degrees. (Image omitted)
What is mAB?
If the chords are parallel, then the arcs they create are congruent.
9x – 10 = 5x + 30
4x = 40
X = 10
AB = 5(10) + 30 = 50 + 30 = 80. Choice (4).
26. The bases of a prism are right trapezoids, as shown in the diagram below. (Image omitted)
Which two edges do not lie in the same plane?
Choice (1), BC and WZ. You cannot find a piece of paper which lines up with both edges. They are skew.
27. In the diagram below, A’B’ is the image of AB under which single transformation? (Image omitted)
It’s flipped over the x-axis and slid back a few spaces. It’s a glide reflection.
28. For which diagram is the statement triangle ABC ~ triangle ADE not always true?
Choice (4), which shows a trapezoid, with A as the midpoint of the diagonals. Triangles ABC and ADE share no angles and have none which must be congruent. Therefore, they are not always similar.