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divided alphabet categories I’veI’ve divided the the alphabet intointo fivefive categories as as shown below. identify category shown below. CanCan youyou identify the the category to to which each letter of the word BRAIN belongs? which each letter of the word BRAIN belongs? Give a brief explanation for your choice. Find another five letter word whose letters each belong in different categories. (1) M T U V W Y (2) C D E K (3) S Z (4) H O X (5) F G J L P Q Midterm Exam 24 students took the midterm Mean = 120/150 = 80% Median = 133.5/150 = 89% Questions 3b and 3c refer to the third Sketchpad construction that you submitted electronically. The diagram and information appear below. D C E ABCD is a rhombus Point P is any point on diagonal AC that is closer to point A than it is to point C P PF AB EP AC A F B 3b. A high school geometry student proved that the two smaller triangles in the diagram are congruent using the following argument: 1. EAP FAP (The diagonals of a rhombus bisect the angles of the rhombus.) 2. EPA PFA (They are both congruent right angles.) 3. AP AP (Reflexive postulate - both triangles have this side in common.) 3. The two triangles are congruent by AAS. Is this proof valid? If not, explain why. 3c. The same student then concluded that AEP and APF are congruent. Do you agree that these two angles are congruent? Explain your answer. MATH 3395 Fall, 2014 MIDTERM E 87 7. In the diagram, AE is parallel to JB . What is the measure of J? Briefly explain how you obtained your answer. J 62 A B C D MATH 3395 Fall, 2014 MIDTERM 10. In the diagram, two pairs of segments are parallel, as indicated. What is the sum of the measures of 1, 2, and 3? 1 43 2 3 A 11. In the diagram, M is the midpoint of BC and AME is a right angle. What is the sum of the measures of A, B, C, D, and E? Explain how you obtained your answer. E D B M 12. The measure of one of the angles of a rhombus is 60 and the length of a side is 10 inches. What is the length of the shorter diagonal of the rhombus? Draw a diagram and explain how you obtained your answer. C AME is a right angle. What is the sum of the measures of A, B, C, D, and E? Explain how you obtained your answer. E D B M 12. The measure of one of the angles of a rhombus is 60 and the length of a side is 10 inches. What is the length of the shorter diagonal of the rhombus? Draw a diagram and explain how you obtained your answer. C In each diagram below (questions 15 – 18), information is marked. State whether two triangles in each diagram can be proven congruent based on the information given and valid deductions that can be made from it. If they can be proven congruent, state the congruence theorem that applies. 15. 16. 70 110 f) True or False: The measure of DEC is equal to the sum of the measures of DAC and DCE. Justify your answer. divided alphabet categories I’veI’ve divided the the alphabet intointo fivefive categories as as shown below. identify category shown below. CanCan youyou identify the the category to to which each letter of the word BRAIN belongs? which each letter of the word BRAIN belongs? Give a brief explanation for your choice. Find another five letter word whose letters each belong in different categories. (1) M T U V W Y A (2) C D E K B (3) S Z N Vertical line symmetry Horizontal line symmetry 180° rotational symmetry (4) H O X I Horizontal and vertical line symmetry, and 180° rotational symmetry (5) F G J L P Q R No symmetry Transformations • Reflections • Rotations These transformations are isometries. • Translations • Dilations An isometry is a transformation in which the original figure and its image are congruent. Thus isometries preserve distance and angle measure. Reflections If A is the reflection image of point A over line l, then l is the perpendicular bisector of line segment AA A l A The Burning House Problem A man is walking in an open field some distance from his house. It’s a beautiful day and he is carrying an empty bucket with him to collect berries. Before long, he turns around and, to his horror, sees that his house in on fire. Without wasting a moment, he runs to a nearby river (which runs in a straight line from east to west) to fill the bucket with water so that he can run to his house to throw water on the fire. Naturally, he wants to do this as quickly as possible. Describe how to construct the point on the river bank to which he should run in order to minimize his total running distance (and time). H M M H M M Q On a billiard table, a player must hit ball A into pocket B without touching the other three balls shown. Where should the player aim? B Reflection image of point A. B B Reflection image of point B. B B Using Geometer’s Sketchpad, construct a rhombus (not a square) by using only the 2 basic draw tools (point and line segment) and reflections. The figure should remain a rhombus when points are dragged. MATH 3395 GROUP PROBLEM SOLVING PROJECT This Group Problem Solving Project has two parts. Part I - consists of ten (10) challenging geometry problems. Each requires that a conjecture be made and/or verified by using Geometer’s Sketchpad, and all but one require proof of the conjecture. In order to receive full credit, your group must complete six (6) of the problems. Part II – An article taken from the Mathematics Teacher (a publication of the National Council of Teachers of Mathematics). In order to receive full credit, your group must read the article and answer the questions relating to it. These questions require the use of Geometer’s Sketchpad. This project is worth a total of 70 points. Regular Polygon – A polygon with all sides congruent and all angles congruent. Sum of the interior angles = (n 2)180 Measure of each interior angle = (n 2)180 n Every regular polygon can be inscribed in a circle. Measure of each central angle = Number of diagonals = n(n 3) 2 360 n 140 40 Rotations Use Geometer’s Sketchpad to construct a regular nonagon. Rotations Use Geometer’s Sketchpad to construct an angle ABC near the left side of the screen. Construct a line segment DE somewhere else on the screen. Use Sketchpad to construct an angle congruent to BAC and having segment DE as one of its sides. B E C A D Translations - Often called glide or slide transformations Use Geometer’s Sketchpad to construct an acute triangle ABC near the lower left of the screen. Construct point P in the center of your screen. Use Sketchpad to construct a triangle congruent to ABC and having point P as one of its vertices. P B C A If you don’t complete these in class, complete as part of tonight’s homework. HW problem 1: Use Geometer’s Sketchpad to construct a rectangle whose side lengths are in the ratio of 2:1 without using the perpendicular, parallel, or midpoint options in the construct menu, and without constructing any circles. HW Problem 2: Construct the letter A using Geometer’s Sketchpad. A Your must be perfectly vertical and symmetric. In other words, it cannot look like this or this . HW Problem 3: Construct a line segment AB using Geometer’s Sketchpad. Without changing line segment AB in any way, construct a rhombus (not a square) so that AB is one diagonal of the rhombus.