Winter 2016 Math 213 Final Exam. Points Possible. Subtotal 100. Total 100

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1 Winter 2016 Math 213 Final Exam Name Instructions: Show ALL work. Simplify wherever possible. Clearly indicate your final answer. Problem Number Points Possible Score Subtotal 100 Extra credit 10 Total 100

2 1) Which shape(s) am I describing below? Be as specific as possible. For example, instead of just saying rectangle, say non-square rectangle, if that is what you mean. Use the least broad name possible; so even though a rhombus is a quadrilateral, if the shape described is a rhombus, don t call it a quadrilateral. a) I have 4 sides, none parallel and my two diagonals are perpendicular. b) I have 3 sides, and two congruent interior angles that sum to 80 degrees. c) I am a regular polygon with a central angle of 40 degrees (you don t need to know the name, just the number of sides). d) I am a regular polygon with an internal angle of 160 degrees (again, you don t need to know the name, just the number of sides). e) I have four sides and two different pair of opposite, congruent angles.

3 2) Measurement a) Find the area of a regular hexagon (6 sides) with side length 2. Hint: Divide it into triangles. b) Find the surface area of a right regular pyramid with a height of 4 cm, and a square base with side lengths of 6 cm. Hint: The given height is not the slant height. c) What is the volume of the largest cylinder that can fit inside a cube of volume 8 cm 3? Hint: The base of the cylinder will lie flat against one side of the cube.

4 3) Similarity and Congruence a) In the picture below, ABCD is a square. Don t assume anything that isn t labeled. For example, AE appears to have similar length to CF, but don t assume they are equal unless you can prove it. i) Find a triangle similar to DEF. Justify why it is similar. List the vertices in corresponding order. ii) Assuming you know that the ratio of DE to EF is 2 3, and you know that the square has an area of 16 cm 2, determine the length of side BE using similar triangles. b) Consider triangles ADE and BCE pictured below. Although the triangles are not necessarily drawn to scale, line segments AD and BC are parallel, and E is a point on the line segment CD and on the line segment AB. i) Is the above enough information to ensure that the triangles are congruent? Why or why not? ii) If the triangles are congruent, does that guarantee that E is the midpoint of AB? Why or why not? iii) If the lengths of AD and BC are equal, does that guarantee that the triangles are congruent? Why or Why not?

5 4) Isometries a) Triangle ABC below was transformed with a rotation isometry. i) Explain how you know that the isometry was not a translation, not a reflection and not a glide reflection. ii) Construct (using a compass and straight-edge) the center of rotation, then label the angle of rotation. Bullet-point the steps that you take (you don t have to bullet point all of the details, just say stuff like construct a line perpendicular to AB at point C. Your sketch will show the details).

6 b) Below are two right triangles. ABC was transformed with an isometry to produce DEF. i) Explain how you know that the isometry was neither a rotation nor a translation. ii) Use algebra (slopes, midpoints and the like) instead of construction, to sketch the axis of reflection. Then, determine the equation of this line. Show your work. iii) What type of isometry transforms ABC to DEF? Describe this isometry completely.

7 Extra Credit: Come up with the formula for the area of a circle by following the steps below. It will be useful to know the ratio of the circle s diameter to its circumference. Take a circle with radius r and (pretend to) cut it into eight congruent pieces. Put the pieces together to form two triangles. The sides of the shape are likely to be curved (scalloped), but you would probably not detect that curvature if there were millions of pieces, not just eight. So, pretend that the curved sides are actually straight and derive an expression for the total area of the two triangles. Show that this expression yields the area formula of a circle.

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