2. If C is the midpoint of AB and B is the midpoint of AE, can you say that the measure of AC is 1/4 the measure of AE?

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1 MATH Midterm Exam 2 Practice Exam Solutions 1. Show two rays in the same plane that intersect at more than one point. Rays AB and BA intersect at all points from A to B. 2. If C is the midpoint of AB and B is the midpoint of AE, can you say that the measure of AC is 1/4 the measure of AE? Yes. Since C is the midpoint of AB, they are all collinear and AC = CB. Since B is the midpoint of AE, A, E and B are collinear and AB = BE. Therefore, A, B, C, and E are all collinear and AC = 1 4 AE. 3. Draw a cube and label the eight vertices. (a) Name two line segments in your cube that are in the same plane. AB and CD are an example of two line segments in the same plane. (b) Name two line segments in your cube that are not in the same plane. AB and EH are an example of two line segments that are not in the same plane. (c) Are there any complementary angles? If so, name one pair. If not, state why not. Not in the lines drawn, but yes there are some in the lines that are not seen. For example, ABD and DBC are complementary (d) Are there any supplementary angles? If so, name one pair. If not, state why not. Yes, for example ABC and DCG are supplmentary. 4. Draw a triangle that is both isosceles and acute. Isosceles because AB = AC, and acute because all angles are less then 90 degrees. 5. Draw an obtuse triangle and construct the median from the largest angle. The median (dashed line) goes from the vertex to the midpoint of the opposite side. 6. Draw an equilateral triangle. Construct one of the medians. What can you say about the two triangles that are created?

2 ABD is congruent to ACD. 7. Draw a triangle in which an altitude is outside the triangle. Altitudes go from a vertex perpendicular to the opposite side. Any obtuse triangle will have an altitude that is outside of the triangle. 8. Is it possible for a triangle to be both obtuse and scalene? Explain your answer. Yes, this is an example because all sides are different lengths (scalene) and one angle is greater than 90 degrees (obtuse). 9. Construct a Venn diagram to represent the relationship between trapezoids and rectangles. Explain or construct examples to justify your diagram. Trapezoids have one pair of parallel sides. Rectangles have all 90 degree angles, which means they have 2 pairs of parallel sides. Therefore, all rectangles are trapezoids. 10. Draw a concave quadrilateral with exactly one right angle. Concave because it caves in, one 90 degree angle. 11. Draw a quadrilateral with exactly one right angle. Same pic as above would work, or this one works as well. 12. How many faces does a hexagonal pyramid have? How many edges? How many vertices? A hexagonal pyramid has a six-sided base up to an apex. It has 7 faces (base plus 6 sides), 12 edges (6 on base, 6 up to vertex), and 7 vertices (6 on base plus apex). 13. Draw a net for a right triangular prism. A right triangules prism has a triangle base and top and lateral faces that are rectangles.

3 14. Consider the following two possible nets. For each one, answer the following: Can the net be used to construct a polyhedron? If so, draw or describe the resulting shape. If not, state why not. The first one cannot because the two top shapes don t cover the top and bottom openings, it s like a cereal box without a bottom. The second one does work, it makes something kind of like a barn. 15. Is there an angle for which a rotation clockwise and a rotation counter-clockwise will result in the same image? If not, why not? If so, what is the angle? Yes, 180 degrees is the same rotation clockwise as it is counterclockwise, regardless of the figure being rotated. 16. Draw the image of the figure below if it is rotated 90 degrees clockwise about the point R. Draw the image if the original is rotated 90 degrees clockwise about the point S. Compare the two images. Are they the same? Same orientation of the shape, but different positions. 17. What single translation would produce the same image as the composition of the translation vector 1 followed by the translation vector 2? Vector 1 shifts a figure 2 units up and 4 units right. Vector 2 shifts a figure 3 units down and 2 units left. Together, the combined translation is 1 unit down and 2 units right. 18. The coordinates of the corners of a quadrilateral are (2,3), (2,6), (5,3), (5,7). What are the coordinates of the corners if the figure is reflected over the line x = 6? What are the coordinates of the corners if the original figure is rotated 90 degrees clockwise about the point (2,3)?

4 Reflection has points (7,3), (7,7), (10,6), and (10,3) Rotation has points (2,0), (6,0), (5,3), and (2,3) 19. Draw a square with sides of 2 units and lower left corner at point A (2,2). What will the coordinates of the upper left corner be? Rotate the figure 90 degrees counter-clockwise about point A and then translate it 2 units left and 3 units up. What will the coordinates of the square be after these transformations? Upper left corner is (2,4). The new coordinates after the transformations are (0,3), (2,3), (0,5), and (2,5). 20. Define the term congruence transformation and give an example. A congruence transformation is a mapping of a figure to a new figure that is congruent to the original. Translations, rotations, and reflections are all congruence transformations. 21. Terry said that he moved triangle A into the same position as triangle B in exactly two moves. Which two changes did Terry make to triangle A? Triangle B is shifted right and horizontally reflected from A. (this is not the only option for a description) 22. How many lines of symmetry does an equilateral triangle have? Does a regular pentagon have rotational symmetry? If so, where is the center of rotation, and what is/are the angle(s) of rotation? Yes, the center of rotation is the center of the pentagon. The angles are 72, 144, 216, 288, and 360 degrees. 24. Which of the following does the arrangement of stars on the American flag have: vertical line symmetry, horizontal line symmetry, rotational symmetry, diagonal line symmetry? Vertical line symmetry through the center, no diagonal or horizontal line symmetry and no rotational symmetry. 25. Which of the digits 0-9 have line symmetry? Vertical line symmetry in the digits 0, 1 (if only drawn as a straight line), and 8. Horizontal line symmetry in the digits 0, 1 (if only a straight line), 3, 8

5 26. Does a prism have rotational symmetry? A prism will only have rotational symmetry if it is a right prism and the base has rotational symmetry. 27. Explain why an octagon used along with a square will tessellate even though an octagon alone will not. A regular octagon has angles of 135 degrees and so cannot tessellate by itself. However, two octagon angles plus one angle from a square gives 360 degrees total and so they could tessellate together. 28. Which of the following, if you traced around them, will tessellate the plane: nickel, dime, quarter, half-dollar, dollar bill? The dollar bill is the only one that will tessellate the plan because it is a rectangle and the others are circles and so cannot fill in all the gaps. 29. Will the figures below tessellate? The first figure is a quadrilateral and all quadrilaterals tessellate. The second and third figures will tessellate assuming the angles and shapes all meet up as they appear to do.