Log1 Contest Round 2 Theta Geometry. 4 points each 1 What is the area of an isosceles right triangle with legs of length 3?

Transcription

1 Log Contet Round Theta Geometry Name: 4 point each What i the area of an iocele right triangle with leg of length? What i perimeter of a regular heptagon with ide of length 8? Matt built a foot concrete walkway around hi circular garden. If hi garden i 6 feet acro in diameter, what i the area of the walkway in quare feet? 4 How many diagonal are there in a regular nonagon? 5 In a regular quadrilateral, what i the ratio of the diagonal length to it ide length? 5 point each 6 How many point of interection are poible in a plane figure that conit of three congruent but ditinct circle? 7 What i the meaure of the maller angle in degree created by the two hand of an analog clock face at 0:0? 8 What i the area of a triangle with ide of length 7, 8, and 9? 9 An ant walk an elliptical path that at it widet major ai meaure 8 inche and it minor ai i 4 inche. What i the area of the region that the ant make? 0 A phere that ha a radiu of 4 cm i cut by a plane that i cm from the center of the phere at it cloet point. What i the area of the circle created by the interection? 6 point each What i the poitive difference between the longet and hortet diagonal that can be formed in a regular heagon of ide length? A maller heagon i incribed in a larger regular heagon by connecting the midpoint of each ide. The maller heagon ha ide length, what i the area of the larger heagon? A conical funnel i placed point down and a pherical marble with radiu 4 mm i dropped in touching the cone in a circle. A larger marble with radiu 6 mm i then dropped in and it touche the firt marble and i tangent to the cone in a circle. A third, till larger, marble i then placed in the cone, touching the econd one and tangent to the cone. What i the radiu of the third marble, in mm? 4 A point i in the interior of a quare ABCD. The point i 0 unit from vertice A and C and unit from verte B. How many unit i the point from verte D? 5 A circle ha equation y 8y 84. What i it area?

2 Log Contet Round Alpha Geometry Name: 4 point each What i the area of an iocele right triangle with leg of length? What i perimeter of a regular heptagon with ide of length 8? Matt built a foot concrete walkway around hi circular garden. If hi garden i 6 feet acro in diameter, what i the area of the walkway in quare feet? 4 How many diagonal are there in a regular icoagon? 5 In a regular quadrilateral, what i the ratio of the diagonal length to it ide length? 5 point each 6 How many point of interection are poible in a plane figure that conit of three congruent but ditinct circle? 7 What i the meaure of the maller angle in degree created by the two hand of an analog clock face at 0:0? 8 What i the area of a triangle with ide of length 7, 8, and 9? 9 An ant walk an elliptical path that at it widet major ai meaure 8 inche and it minor ai i 4 inche. What i the area of the region that the ant make? 0 A phere that ha a radiu of 4 cm i cut by a plane that i cm from the center of the phere at it cloet point. What i the area of the circle created by the interection? 6 point each What i the length of the longet diagonal that can be formed in a regular octagon of ide length? A maller heagon i incribed in a larger regular heagon by connecting the midpoint of each ide. The maller heagon ha ide length, what i the area of the larger heagon? A conical funnel i placed point down and a pherical marble with radiu 4 mm i dropped in touching the cone in a circle. A larger marble with radiu 6 mm i then dropped in and it touche the firt marble and i tangent to the cone in a circle. A third, till larger, marble i then placed in the cone, touching the econd one and tangent to the cone. What i the radiu of the third marble, in mm? 4 A point i in the interior of a quare ABCD. The point i 0 unit from vertice A and C and unit from verte B. How many unit i the point from verte D? 5 A circle ha equation y 8y 84. What i it area?

3 Log Contet Round Mu Geometry Name: 4 point each What i the area of an iocele right triangle with leg of length? What i perimeter of a regular heptagon with ide of length 8? Matt built a foot concrete walkway around hi circular garden. If hi garden i 6 feet acro in diameter, what i the area of the walkway in quare feet? 4 How many diagonal are there in a regular icoagon? 5 What i the ratio of the length of the larget diagonal to the ide of a regular heagon? 5 point each 6 How many point of interection are poible in a plane figure that conit of three congruent but ditinct circle? 7 What i the meaure of the maller angle in degree created by the two hand of an analog clock face at 0:0? 8 What i the area of a triangle with ide of length 7, 8, and 9? 9 An ant walk an elliptical path that at it widet major ai meaure 8 inche and it minor ai i 4 inche. What i the area of the region that the ant make? 0 The lateral urface area of a right circular cone i three time the area of the bae. If the bae area i 6, what i the volume of the cone? 6 point each What i the length of the longet diagonal that can be formed in a regular octagon of ide length? A maller heagon i incribed in a larger regular heagon by connecting the midpoint of each ide. The maller heagon ha ide length, what i the area of the larger heagon? A conical funnel i placed point down and a pherical marble with radiu 4 mm i dropped in touching the cone in a circle. A larger marble with radiu 6 mm i then dropped in and it touche the firt marble and i tangent to the cone in a circle. A third, till larger, marble i then placed in the cone, touching the econd one and tangent to the cone. What i the radiu of the third marble, in mm? 4 What i the area of the pherical triangle drawn on a phere of radiu cm with angle of meaure, in radian, of,,. 5 The curve y and y are defined on the interval [0,]. What i the area between thee curve on that interval?

4 Log Contet Round Geometry Anwer Theta Anwer Alpha Anwer Mu Anwer [ft ] 7 [ft ] 7 [ft ] or : 5 or : 5 or : [ ] 7 5 [ ] 7 5 [ ] [mm] 9 [mm] 9 [mm] [ q. cm]

5 Log Contet Round Geometry Solution Mu Al Th Solution A bh Area of walkway Area of larger circle Area of Garden n n No. of diagonal. Each of the n vertice can be connected to n- other but thi would double count each diagonal : : : : Angle at 0:0 without the added movement of the hour hand 0 0 Movement of hour hand

6 8 8 8 Heron formula: A a b c A i the area, i the emi-perimeter, and the ide length are a, b, and c A A ab a minor ai b major ai A r 4 5 A Lateral area rl Bae r 6 r 6 6 6l 08 6l l 8 h l r 4 V r h longet hortet

7 co 5 c 4 The apothem of the larger heagon i b. Thi make the ide length of the larger heagon 4 /. The area of the heagon um of 6 triangle i then: A apothem perimeter / 64/ 8 / The radii of the marble form a geometric equence. Thi can be een by the conidering the drawing. r, r,, we have r r r r r r r r Cro multiply and implify give: r r r. Thu, the third marble will be 6 6/4 9 mm.

8 4 4 Since the point i equiditant from A and C, it mut be on the BD diagonal. From that, the ide length can be een to 4 and the miing length calculated. What follow i a more general olution. Let v and w be the horizontal and vertical diplacement of the point from verte A. Let the ditance from the point to verte D be d, the ide length, and we get the following 4 equation uing the Pythagorean Formula: d w v w v w v w v Subtract the econd equation from the firt and the third from the fourth. 0 8 d w w From which it follow: 8 d 4 The area i um of the angle - time the radiu quared. A q cm y y i equivalent to 6 7 y. The area i 6 r A 5 The curve cro at, and o the interval mut be broken into two egment d d A