008 009 Log Contest Round Theta Geometry Name: Leave answers in terms of π. Non-integer rational numbers should be given as a reduced fraction. Units are not needed. 4 points each What is the perimeter of a square with a diagonal of units? A trapezoid has an area of 6 square units, a height of 8 units and a base of units. Find the length of the remaining base. Cube A has a volume of 4 cubic units. Cube B has a volume of 7 cubic units. Cube C has a volume of 64 cubic units. Cube D has a volume of 5 cubic units. Find the sum of the surface areas of Cubes A, B, C, and D. 4 Triangle ABC is given such that AB and BC have lengths of 4 units. If CAB has an angle measure of 54, what is the measure, in degrees, of ABC? 5 What is the y-coordinate of the y-intercept of f ( x ) ( x 4)( x + 9)? 5 points each 6 Determine the sum of the interior angles of a regular dodecagon in degrees. 7 CD is tangent to circle P as shown. AB is a diameter of circle P. CD6, BC4. Find the radius of circle P. 8 A pyramid is constructed with a square base and all edges equal to. What is the volume of the pyramid? 9 Find the value of xy, disregarding units. 0 A ship starts at point A and travels 9 km North. It then goes 5 km West and stops. How far, in km, is the ship from point A?
6 points each How many diagonals can be placed in a convex octagon? A circle of radius is tangent to two perpendicular lines. What is the radius of the largest circle that can be placed between the circle and corner and tangent to the lines and the larger circle? x 4 y 5 y x + 7 Determine the area of the shape enclosed by these restrictions. 4 If the volume of a sphere is doubled, what is the ratio of the new radius to the old one in the form A:B? 5 What is the area of this circle?
008 009 Log Contest Round Alpha Geometry Name: Leave answers in terms of π. Non-integer rational numbers should be given as a reduced fraction. Units are not needed. 4 points each What is the perimeter of a square with a diagonal of units? A trapezoid has an area of 6 square units, a height of 8 units and a base of units. Find the length of the remaining base. Cube A has a volume of 4 cubic units. Cube B has a volume of 7 cubic units. Cube C has a volume of 64 cubic units. Cube D has a volume of 5 cubic units. Find the sum of the surface areas of Cubes A, B, C, and D. 4 Triangle ABC is given such that AB and BC have lengths of 4 units. If CAB has an angle measure of 54, what is the measure, in degrees, of ABC? 5 What is the area, in square centimeters, of a regular octagon with sides of length 0 centimeters? 5 points each 6 Determine the sum of the interior angles of a regular dodecagon in degrees. 7 CD is tangent to circle P as shown. AB is a diameter of circle P. CD6, BC4. Find the radius of circle P. 8 A pyramid is constructed with a square base and all edges equal to. What is the volume of the pyramid? 9 Find the value of xy, disregarding units. 0 What is the radius of the following circle? 4x + 4y 6x 64y 5
6 points each How many diagonals can be placed in a convex octagon? A circle of radius is tangent to two perpendicular lines. What is the radius of the largest circle that can be placed between the circle and corner and tangent to the lines and the larger circle? x 4 y 5 y x + 7 Determine the area of the shape enclosed by these restrictions. 4 If the volume of a sphere is doubled, what is the ratio of the new radius to the old one in the form A:B? 5 A circle is completely divided into n distinct sectors. The angles of these sectors form an arithmetic progression. If the smallest angle is, and the largest is 68, determine the value of n.
Name: 008 009 Log Contest Round Mu Geometry Leave answers in terms of π. Non-integer rational numbers should be given as a reduced fraction. Units are not needed. 4 points each What is the perimeter of a square with a diagonal of units? A trapezoid has an area of 6 square units, a height of 8 units and a base of units. Find the length of the remaining base. Cube A has a volume of 4 cubic units. Cube B has a volume of 7 cubic units. Cube C has a volume of 64 cubic units. Cube D has a volume of 5 cubic units. Find the sum of the surface areas of Cubes A, B, C, and D. 4 If the pattern continues, what is the sum of all of the perimeters as the number of triangles approaches? 5 What is the area, in square centimeters, of a regular octagon with sides of length 0 centimeters? 5 points each 6 Determine the sum of the interior angles of a regular dodecagon in degrees. 7 CD is tangent to circle P as shown. AB is a diameter of circle P. CD6, BC4. Find the radius of circle P. 8 A pyramid is constructed with a square base and all edges equal to. What is the volume of the pyramid? 9 What is the volume of when the region between f ( x ) x and g ( x ) x is rotated around the x-axis? 0 What is the radius of the following circle? 4x + 4y 6x 64y 5
6 points each How many diagonals can be placed in a convex octagon? A circle of radius is tangent to two perpendicular lines. What is the radius of the largest circle that can be placed between the circle and corner and tangent to the lines and the larger circle? x 4 y 5 y x + 7 Determine the area of the shape enclosed by these restrictions. 4 A ladder is leaning on the side of a house with the foot 7 feet from the wall and the top 4 feet from the ground. The ladder starts to slip and the bottom moves away from the wall at a constant rate of 6 inches/second. At what rate, in radians per second, is the angle between the ladder and the wall changing when the bottom is 0 feet from the wall? 5 A circle is completely divided into n distinct sectors. The angles of these sectors form an arithmetic progression. If the smallest angle is, and the largest is 68, determine the value of n.
008 009 Log Contest Round Geometry Answers [ ] represent optional units Theta Answers Alpha Answers Mu Answers 4 [units] 4 [units] 4 [units] 6 [units] 6 [units] 6 [units] 594 [ units ] 594 [ units ] 594 [ units ] 4 ABC 7 [ ] 4 ABC 7 [ ] 4 56 [units] 5-6 5 00 + 00 5 00 + 00 [ cm ] [ cm ] 6 800 [ ] 6 800 [ ] 6 800 [ ] 7 Radius of circle P.5 8 4 8 7 Radius of circle P.5 4 8 9 8 9 8 9 0 65 km 0 9 0 9 7 Radius of circle P.5 4 π 5 0 0 0 [ units ] [ units ] 4 : 4 : 4 5 69π [ units ] 5 9 5 9 or + [ units ] [radians/ 0 sec]
007 008 Log Contest Round Applications Solutions Th Al Mu Solution The diagonal of a square is equal to x units, where x equals the side length of the square. Thus the side length is unit, making the perimeter 4 units. A h(b + b ) 6 (8)( + b ) b 6 Surface area of Cube A 6(7) 94 B 6() 54 C 6(4) 96 D 6(5) 50 4 4 94 + 54 + 96 + 50 594 x 80 (54) x 80 08 x 7 4 The individual perimeters of each triangle form a geometric sequence: (56 + 8 + 64 + + 6 + 8 +...) t ( ) r 56 ( ) (5) 56 5 To find the y-intercept, evaluate when x0. b (-4)(9) b -6 5 5 Sketch the octagon and extend the sides to form a square. The area of the octagon is the area of the square minus the area of the four resultant triangles. A ( 0 + 0 ) (5 ) 00 + 00 6 6 6 7 7 7 S 80( n x ) S 80( ) X 800 ( CD) ( AC ) ( BC ) 6 ( AC ) 4 AC 9 9 4.5 8 8 8 The volume of a pyramid is one-third the product of the area of the base times the height. The area of the base is 4. The height can be found by dropping a perpendicular from the top to the base, from there perpendicular to one side of the base and back the top. The hypotenuse is the height of an equilateral triangle with side. This is. The height of the pyramid is then. The volume is 4.
9 9 Set up a proportion to solve for x. x x 9 4 y 80 5 46 y 8 xy 4 8 xy 8 9 y x y x V π ( y y ) dx 0 π 4 ( x x ) dx 0 π ( x x 5 ) 5 0 π ( ) 5 π 5 0 The distances form a -4-5 right triangle with legs of () and 4(), so the hypotenuse/distance from point A is 5(), or 65. 0 0 Complete the square: x + y 4x 6y x 4x + 4 + y 6y + 64 + 4 + 64 8 9 Number of diagonals: n( n ) 8(5) 0 From the center of the larger circle drop a perpendicular to one of the lines and to the center of the smaller circle. Draw a perpendicular from the center of the smaller circle to the first perpendicular. Call the radius of the smaller circle, x. One gets a right triangle with two sides equal to -x and a hypotenuse of +x. ( x ) + ( x ) ( + x ) x 6x + 0 x ± Only the smaller value is between 0 and. Or consider the line drawn from the center of the larger circle (,) through the center of the small circle (x,x) to the origin (intersection of the lines). The whole length is but it can be divided into the radius of the larger circle, smaller circle x, and from the center of the smaller circle to the origin x. So + x + x.
A bh A (8)(8) A 4 4 4π Volume of Sphere: r 8π Double of Sphere: r Radius of Sphere: r Radius of Doubled Sphere: R 4 4π 8π R r R r R r R r R : r : x sinθ 5 dθ dx cosθ dt 5 dt dθ cosθ dt 50 5 dθ 5 dt 50 dθ dt 0
5 An inscribed right triangle s hypotenuse is always the length of the diameter of the circle. Also the triangle is a 5-- right triangle, albeit by a factor of two. r hypotenuse () A πr 69π 5 5 first + last total n average n 60 n 9 + 68