What follows is a portion of the Common Core Geometry exam. Other parts will be posted on other days. Illustrations will be added at a later time when they become available.
June 2016 Geometry Regents, Part II
25. Describe a sequence of transformations that will map triangle ABC onto triangle DEF as shown below.
There are multiple possible answers. One possibility is a reflection over the x-axis, followed by a translation 4 units to the right. (The reverse order would work, too.)
26. Point P is on segment AB such that AP:PB is 4:5. If A has coordinates (4,2), and B has coordinates (22,2), determine and state the coordinates of P.
The distance between the two x-coordinates is 18 units. If AP:PB is 4:5, then the two segments are 4x and 5x in length, and 4x + 5x = 18. So 9x = 18, and x = 2.
4(2) = 8, 4 + 8 = 12. P is at (12, 2).
27. In triangle CED as shown below, points A and B are located on sides CE and ED, respectively. Line segment AB is drawn such that AE = 3.75, AC = 5, EB = 4.5, and BD = 6.
Explain why AB is parallel to CD.
As shown in the illustration below, if you can show that the sides of the smaller triangle and the larger triangle are proportional, then the triangles are similar. If they are similar, then the corresponding angles of the two triangles are congruent.
Angle EAB is congruent to angle ECD and they are corresponding angles on a transversal. Therefore, AB is parallel to CD.
28. Find the value of R that will make the equation sin 73° = cos R true when 0° < R < 90°.
Explain your answer.
The sine of an angle is equal to the cosine of the complementary angle. 90 - 73 = 17.
R = 17.
EDIT Arithmetic error corrected. Minus 1 for me.
29. In the diagram below, Circle 1 has radius 4, while Circle 2 has radius 6.5. Angle A intercepts an
arc of length pi, and angle B intercepts an arc of length 13*pi / 8.
Dominic thinks that angles A and B have the same radian measure. State whether Dominic is correct or not. Explain why.
The measure of angle A is arclength / radius, which is pi / 4 radians.
The measure of angle B is 13(pi)/8 / 6.5, which is (13 pi) / (8*6.5) = (13 pi) / 52 = pi / 4.
Angle A and B have the same measure.
Another approach, which is a little more old-school, is to use circumference to find arclength.
The circumference of a circle is pi*d or 2*pi*r.
The length of an arc of a circle is the circumference times the size of the angle/360.
In the first circle, (A / 360) (2) (4) (pi) = pi
So (A / 360) (2) (4) = 1
Use inverse operations to isolate A, and A = 45.
In the second circle, (B / 360) (2) (6.5) (pi) = 13(pi) / 8
So (B / 360) (2) (6.5) = 13 / 8
Use inverse operations to isolate B, and B = 45.
So A and B have the same measure and that is 45.
30. A ladder leans against a building. The top of the ladder touches the building 10 feet above the ground. The foot of the ladder is 4 feet from the building. Find, to the nearest degree, the angle that the ladder makes with the level ground.
The ladder makes a right triangle. The wall is the opposite side. The ground is the adjacent side. The ladder is the hypotenuse, but we don't know or need to find the length of the ladder. So we have opposite and adjacent, so we are using tan. More specifically, since we are looking for the size of the angle, we need tan-1.
tan(x) = 10/4, so x = tan-1 (10/4) = 68 degrees.
31. In the diagram below, radius OA is drawn in circle 0. Using a compass and a straightedge, construct a line tangent to circle 0 at point A. [Leave all construction marks.]
Sorry, but I'm still not good with handling constructions electronically.
Use the straightedge to extend the radius. From the point A, measure off two points on other side. Then construct a perpendicular bisector. Let's call these first two marks B and C. Go to B and make an arc above and below the line. Do the same at C so that you make two X's. Draw the perpendicular bisector, it will be tangent to the circle.
End of Part II