Geometry Team #1 FAMAT Regional February
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1 Geometry Team #1 FAMAT Regional February A. Find the area of a triangle with semi perimeter 13 and two sides having lengths 10 and 9. B. In DPQR, ÐQis obtuse, mð P= 45, PR= 10, PQ= 3. Find the area of the triangle. C. Find the length of the altitude to the longest side of a triangle with vertices 2,1, 3,4, 6,3. ( ) ( ) ( ) D. The altitude drawn to the base of an isosceles triangle is 8 and the perimeter is 32. Find the area of the triangle. Geometry Team #2 FAMAT Regional February A. DABCis isosceles and the perimeter of the triangle is less than 45. When AB = 10, BC = x + 7, AC = 2x -8, give the name of the segment that is the base of the triangle. B. Find the number of distinct scalene triangles having all sides of integral lengths and perimeter less than 13. C. Two sides of a triangle are 10 and 22. Find the average of the possible integral values of the third side. D. Find the measure of an acute angle if twice the measure of its supplement is 24 more than five times the measure of its complement. Geometry Team #3 FAMAT Regional February Given a regular hexagon with apothem having length 12. A. Find the perimeter of the hexagon. B. Find the area of the hexagon. C. Find the area of the circumscribed circle. D. Find the circumference of the inscribed circle.
2 Geometry Team #4 FAMAT Regional February A. ACand ADare tangent segments of a circle with point A in the exterior of the circle, points C and D are on the circle, point B is on the circle on major CD!. Find the mcbd! when mð A= 92. B. Find the length of a 40 arc in a circle with area 81 p. C. Two non-intersecting circles have radii 5 and 17 and a common external tangent segment of length 16. Find the distance between the centers of the circles. D. The numerical area of a circle is 6 times the numerical circumference of the circle. Find the area of the circle. Geometry Team #5 FAMAT Regional February A. In DABCwith right angle at C, D is a point in the interior of the triangle such that ADis the angle bisector of ÐCAB and BDis the angle bisector of ÐABC. Find the mðadb. B. In D ABC with right angle at C, D is a point on AC, BC = 6, AB = 12, AD = 4 3. Find the value of BD + CD. C. DABCis a right triangle with right angle at B. D is a point on ABsuch that CD is an angle bisector of ÐACB, AC = 25, CB = 15. Find the length of CD. D. Given DABCwith AB = 10, mð B = 45, BC = 3. Find the area of the triangle.
3 Geometry Team #6 FAMAT Regional February A. The diagonals of a rhombus are 15 and 20. Find the perimeter of the rhombus. B. Find the area of a rhombus with side length 12 and the measure of one angle of the rhombus is 150. C. The area of a rhombus is 348 and one diagonal is 24. Find the length of a side. D. A circle is inscribed in a rhombus. Find the area of the region inside the rhombus and outside the circle if the diagonals of the rhombus are 30 and 40. Geometry Team #7 FAMAT Regional February Given: DABCwith points Dand Eon ABand AC, respectively with DE BC. A. Find AE + BC + BD when AD = 12, AB = 20, DE = 15, AC = 30. A B. Find EC + ABwhen AD = 18, AE = 26, DE = 24, BC = 36. C. Find DE + AC when AD = 6, DB = 4, AE = 9, BC = 15. D. Find BD + DE when AB = 33, AE = 24, EC = 20, BC = 50. D B E C Geometry Team #8 FAMAT Regional February A. Find the area of a regular pentagon with a perimeter of 50 and radius length 3 3. B. Find the perimeter of a regular polygon with sides of length 13 and interior angle measure of 156. C. The area of a circle inscribed in a regular hexagon is 100p. Find the perimeter of the hexagon. D. Find the area of a regular hexagon with the distance between the parallel sides is 36.
4 Geometry Team #9 FAMAT Regional February A. In trapezoid ABCD with AB CD, find the area of the trapezoid ABCDwhen AB = 8, DC = 15, BC = 10, mð C = 30. B. A square and a rectangle have equal areas. The rectangle has one side 16 and a diagonal of 20. Find the length of a side of the square. C. Find the area of polygon ABCDEF with vertices A -2,2, B 2,7, C 8,0, D 4, -8, E -2,-6. ( ) ( ) ( ) ( ) ( ) D. A rhombus is given with one diagonal twice the length of the other diagonal. Express the side of the rhombus in terms of k, where k is the area of the rhombus. Geometry Team #10 FAMAT Regional February A. An isosceles trapezoid with bases 12 and 16 is inscribed in a circle of radius 10. The center of the circle is in the interior of the trapezoid. Find the area of the trapezoid. B. A regular pentagon is inscribed in a circle. Find the measure of the angle formed by a side of the pentagon and a line tangent to the circle at one of the vertices of the pentagon. C. Give the ratio of the apothem of a regular hexagon to the radius of the circumscribed circle. D. Quadrilateral ABCD is circumscribed about circle O. If find the perimeter of ABCD. AB = 6, DC = 11,
5 Geometry Team #11 FAMAT Regional February Given: Circles P and O with tangent segment RT = 48, OT = 50, OP = 25. TS. R and S are points of tangency. A. Find the length of SP. B. Find the area of DSTP. C. Find the perimeter of quadrilateral RSPO. D. Find the area of quadrilateral RSPO. Geometry Team #12 FAMAT Regional February A. The area of the larger of two similar polygons is 64. When the corresponding sides of the two polygons are 6 and 12, find the area of the smaller polygon. B. The area of trapezoid ABCD is 90. Base AB has length 8 and base CD has length 12. DAand CB are extended until they meet at point E. Find the area of triangle EAB. C. Find the circumference of a circle in which a 48 unit chord is 8 units closer to the center than a 40 unit chord. D. The front wheels of a wagon are 2p ft in circumference and the back wheels are 3p ft in circumference. Find the distance in feet that the wagon has traveled when the front wheels have made 10 more revolutions than the back wheel.
6 Geometry Team #13 FAMAT Regional February A. Find mðawhen mbd! = 80. B. Find mad! when mð ADM = 75. C. Find mac! when mbd! = 80,m BND = 81. D. Find the length of CDwhen AN = 12, BN = 6, CN = 8. Geometry Team #14 FAMAT Regional February A. Point B divides APinto segments ABand BP. If AP = 75and find the length of BP. B. Find the area of the right triangle with hypotenuse having length 113and legs having a sum of 15. C. A rhombus contains a 120 angle. Find the ratio of the length of the shorter diagonal to the length of the longer diagonal. D. In D ABC, CA, mð B = 2x -36, mð C = x + 2. Find the mða. AB 2, BP = 3
7 Geometry Team #15 FAMAT Regional February A. ABChas vertices at A 6,2, B -4,6, C -6,-4. Find the length of the D ( ) ( ) ( ) median to BC. B. Find the area of a circle when the endpoints of the diameter are 3,1 and (- ) ( ) C. T is the midpoint of the segment with endpoints 7,2 and 5,4. S is the x-intercept of the line 2x+ y= 7. Find the length of TS. (- ) ( - ) 0, 5. D. Find the circumference of the circle with equation: 2 2 x y x y = 0.
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