Florida Association of Mu Alpha Theta January 2017 Geometry Individual Solutions

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1 Geometry Individual Solutions lorida ssociation of u lpha Theta January 017 Regional lorida ssociation of u lpha Theta January 017 Geometry Individual Solutions Individual nswer Key uthor: Sayeed Tasnim, Seminole High School lass of 011 1

2 Geometry Individual Solutions lorida ssociation of u lpha Theta January 017 Regional 1. () If I get guacamole in my hipotle burrito bowl, I do not get chicken in my hipotle burrito bowl. This is the contrapositive and the contrapositive of a valid if-then statement is always true.. () (3,3). There are three possible points to make a parallelogram, each opposite one of the side lengths. To determine the point opposite side, move starting from point in a direction parallel to and travelling the same distance. to is 9 units to the right and 11 units down. oving the same amounts from, the resulting point is (17, 9). lternatively one could move starting from point and travel parallel to with the same distance to yield the same point. Repeating the same process for all of the sides, the three points are = (17, 9), = ( 13, 1), and = ( 1, 13). The sum of the coordinates of these 3 points is (3, 3). However, if one notices that the three possible points due to the parallelism and equal lengths produces many nice parallelograms, it can be shown that triangle is the medial triangle of. The sum of the coordinates of a triangle is equivalent to the coordinate of the centroid (which is the average of the coordinates) multiplied by 3. The centroids of a triangle and its medial triangle coincide. Therefore, one can take the average of the original coordinates of which is (1, 1) and multiply it by 3 to get the answer without having to determine,, and. 3. () 4. onsider the diagram below. = 5, = 1 3, = = = 1 15, = () 3 0. Given that the hexagon has a side length of 4 for simplicity. irst Solution: Reflect over. Let the reflection of points and be and respectively. Since by symmetry (or angle chasing), lies on and is the midpoint of. Let intersect and at Q and N respectively. Q N Since, we have similar isosceles triangles. = 8 by dividing the hexagon into six equilateral triangles. Q = since Q is a right triangle. N = 1 since N is also a right triangle. N = N = 8 1 = 7. Therefore, the ratio of similarity for is Q : N or : 7. Hence, = 5. Since with a ratio of : 1, = 1. gain by, = 7 = 1 7. y symmetry, the remaining 6 7 is split evenly between and so = 3 7, = 4 7, so = 3 4. inally, = 3 0. The motivation for this solution is to consider finding a good line of reflection since is not a very nice piece of the figure. Since point is symmetric about and is an isosceles triangle, the line of symmetry, that is, is a good candidate. Second Solution: Let O be the center of the hexagon. raw line O. Let Q be the foot of the altitude of point to O and draw the perpendicular O from O. Note that O.

3 Geometry Individual Solutions lorida ssociation of u lpha Theta January 017 Regional Q N O y parallel lines, N. Thus, we determine the ratio of similarity which yields. y drawing the six congruent equilateral triangles, Q = O and Q O. QN = NO since they are both perpendicular altitudes to parallel lines. lso, NQ = NO by parallel lines. y S theorem, QN = ON. Since O = 4 and Q is a midpoint (by viewing equilateral triangle O), Q = QO = and QN = NO = 1 by QN = ON congruence. We have N = O NO = 3. The ratio of similarity for N is N : which is 3 : 4 so = 3 4. To determine N = 1 by QN = ON congruence, so = 7. Therefore, = 5. inally, = 3 0., note that N = 4 7 and To motivate this solution, seek to find similar triangles. Since none can be seen from the initial diagram, search for good candidates of parallel lines to draw to force similar triangles. Line segment O is a good candidate since it is parallel to. 5. (). Reflecting a point about the y-axis negates the x-coordinate. Reflecting a point about the line x = y reverses the x and the y coordinates. Therefore, reflecting (, 3) about the y-axis results in (, 3) and reflecting that point about the line y = x results in (3, ). 6. () 4. y the angle bisector, =. y opposite interior angles due to a parallelogram, =. y transitivity, = and therefore triangle is isosceles. This implies that = = 6. Thus, = = 10 6 = () 70. Since and are isosceles triangles, we have that m = 180 m and m = 180 m. m + m + m = 180 since they form a straight angle so m = 180 m +m m m = = m +m = 140 = 70 where we used the fact that the sum of the measures of the angles of triangle is

4 Geometry Individual Solutions lorida ssociation of u lpha Theta January 017 Regional () 18. Let a be the length of one of the congruent legs of the isosceles triangle and let b be the length of the base. We have a + b = 40. y the triangle inequality, b < a + a = a. Therefore, b = b + b < a + b = 40 so b < 0. However, if b = 19, then a = 1 which is not an integer. Therefore, b = 18 is the maximum length of the base. 9. () 1. One of the coordinates can be 0 and the other coordinate can be ±5 for 4 choices. One of the coordinates can be ±3 and the other coordinate can be ±4 for a total of 8 choices. Therefore, there are 1 total choices. 10. () 96. The 48-by-7 rectangle can be divided into -by-3 rectangles arranged in a 4-by-4 fashion. The diagonal of the 48-by-7 rectangle can be considered as the concatenation of 4 diagonals of -by-3 rectangles. The diagonal of each -by-3 rectangle passes through 4 unit squares. Therefore, the diagonal of the 48-by-7 passes through 4 4 = 96 unit squares. 11. () 1. Since by parallelogram, G = = and G = =. y similarity, G G. Since is the midpoint of and =, = = 1 so the ratio of similarity for triangles G G is 1 :. Therefore, G = 6 and G = 1. G lso note that this effectively proves the centroid property that the centroid splits medians into the ratio 1 :. 1. () Reflex. y definition, a reflex angle is an angle that a measure greater than () diagonal splits the quadrilateral into two congruent triangles. If a diagonal splits a quadrilateral into two congruent triangles, the quadrilateral may be a kite that is not a parallelogram. Hence, it is not a valid way to identify a parallelogram. The remaining answer choices are all valid methods of identifying a parallelogram. 14. () 45. Let m = m = m = x. y right triangle, m = (90 3x). The the sum of of the angles in triangle, x + (90 3x) = 180. Therefore, x = 15 and m = 3x = 45. 4

5 Geometry Individual Solutions lorida ssociation of u lpha Theta January 017 Regional () cute. Using the triangle classification test, > 7 so the triangle is acute. 16. () If Luffy does not find One iece, then he will not be the pirate king. The inverse of an if-then statement negates the antecedent and the consequent of the statement. 17. () 35. m = 10, m G = 40, and m G I = 90 imply that m I = 110. Since triangle I isosceles with = I, m I = 35. G H I 18. () 4. The hypotenuse is twice the shortest leg in a right triangle and the shortest leg is opposite the 30 angle. 19. () 5. side of a triangle can be characterized by how many sides of the octagon it spans. or example, in octagon GH, a triangle with side spans one side of the octagon, side spans two sides of the octagon, etc. triangle can be characterized by a triple of positive integers that sum to 8 corresponding to the number of sides of the octagon that each side of the triangle spans. numerating, there are five such triples: (1, 1, 6), (1,, 5), (1, 3, 4), (,, 4), (, 3, 3). xamples of these triangles are,,,, and. ll other triangles are congruent to one of these five. 0. () 180. This is an example of the angles in a star formula which is (n 4) 180 where n is the number of points in the star. onsider an n pointed star which is formed by extending the sides of an n-gon. The sum of the angles of the points of the star is equal to the sum of the angles of the triangles of the star (n 180 ) subtracted by the angles that are adjacent to the n-gon. The angles adjacent to the n-gon are exterior angles and each exterior angle is counted twice. The sum of the exterior angles of an n-gon is 360 and since it is counted twice, 70 is subtracted. Therefore, the sum of the angles is n = (n 4) () 1. n angle and its supplement have measures that sum to 180. Therefore 3x x + 13 = 180 x = 18. n angle of measure (4x 3) = 69. The complement of 69 is 1.. () 4. y parallel lines and alternate interior angles, = and =. lso, since = by opposite angles, by similarity. Therefore, we have that the ratio of the heights of triangle to triangle is 6 : 8. Therefore, the height of triangle is 4. 5

6 Geometry Individual Solutions lorida ssociation of u lpha Theta January 017 Regional 3. () 119. Using the ythagorean theorem, + = so 5 + = 1 and = () II only. or the first quadrilateral, if the sum of the lengths of the three shortest sides is less than the length of the fourth longest side, the quadrilateral is impossible to make. Since < 41, this quadrilateral is impossible. ormally, this can be shown by drawing a diagonal of the quadrilateral and applying the triangle inequality twice. Intuitively, this cam be seen by arranging the three smallest side lengths to be aligned in the same direction. The fourth side cannot exceed the sum of the lengths as the three smallest side lengths cannot extend past such a length. trapezoid is possible if and only if a triangle is possible with the legs of the trapezoid and the difference in the bases of the trapezoid. This is seen by translating the legs of the trapezoid together along the parallel direction of the bases until the two legs share a common vertex. When this occurs, a triangle is formed where two of the side lengths are the legs and the third side length is the difference of the bases. triangle with side lengths 5, 8, and 40 8 = 1 is possible because > 1 and 8 5 < 1 by the triangle inequality. 5. () 90. Quadrilateral is a kite and the diagonals of a kite are perpendicular. 6. () 16. The resulting figure is an isosceles trapezoid with a base of 6, a base of, and two congruent legs of 6 = 4. The perimeter is = () 90. Listing the vertex angles first, triangles,, and are all isosceles. Let m = x. m = x and m = x by an exterior angle of a triangle. m = (180 4x) with m = x and the straight angle at yields m = 3x. m + m = x + 3x = 60 since m = 10. Therefore, x = 15 and m = (10 x) = () 16. Using parallel line theorems, place the three angles given by x, 3x, and 5x on a straight line such that x + 3x + 5x = 180. Thus, we have x = 18. Similarly, y = x + 5x = 7x = () 45. Since = K, =, and = K, by S theorem, = K so K =. Therefore, K = K = 54 9 = 5. K = by congruence and = 0 by ythagorean theorem on triangle. inally, K + K = = 45. 6

7 Geometry Individual Solutions lorida ssociation of u lpha Theta January 017 Regional K 30. () 10. Since is right and m + m = 90 (as the angles of triangle sum to 180 ), m = 90 m = m. Therefore, and = 7 = 1 7. Solving for, = 7 =