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This Lesson (OVERVIEW of my additional lessons on miscellaneous advanced Geometry problems) was created by by ikleyn(52957)
  About ikleyn: OVERVIEW of my additional lessons on miscellaneous advanced Geometry problemsMy additional lessons on miscellaneous advanced Geometry problems in this site are - Find the rate of moving of the tip of a shadow - A radio transmitter accessibility area - Miscellaneous geometric problems - Miscellaneous problems on parallelograms - Remarkable properties of triangles into which diagonals divide a quadrilateral - A trapezoid divided in four triangles by its diagonals - A problem on a regular heptagon - The area of a regular octagon - The fraction of the area of a regular octagon - Try to solve these nice Geometry problems ! - Find the angle between sides of folded triangle - A problem on three spheres - A sphere placed in an inverted cone - An upper level Geometry problem on special (15°,30°,135°)-triangle - A great Math Olympiad level Geometry problem - Nice geometry problem of a Math Olympiad level List of lessons with short annotationsFind the rate of moving of the tip of a shadow Problem 1. A street light is at the top of a 18 ft tall pole. A man 6 ft tall walks away from the pole with a speed of 8 ft/sec along a straight path. How fast is the tip a shadow moving when he is 40 ft from the base of the pole? A radio transmitter accessibility area Problem 1. A small radio transmitter broadcasts in a 21 mile radius. If you drive along a straight line from a city 25 miles north of the transmitter to a second city 29 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter? Miscellaneous geometric problems Problem 1. What is the distance between the tips of the minute hand and the hour hand of a clock at 1:35 pm, where the length of the minute hand is 14 cm and the length of the hour hand is 9 cm. Problem 2. Two circles with the centers at A and B intersect and share a common chord CD. The radius of circle A is 10 in, the radius of circle B is 16 in. The distance between centers is 22 in. Find CD. Problem 3. A paper cone of circular base has the diameter of 10 cm and the height of 12 cm. It is flattened out into the sector of a circle. What is the angle of the sector? Problem 4. Gibb’s Hill Lighthouse, Southampton, Bermuda, in operation since 1846, stands 117 feet high on a hill 245 feet high, so its beam of light is 362 feet above sea level. Find the distance from the top of the lighthouse to the horizon. Problem 5. Regular hexagon ABCDEF has sides of length 2. The point P is the midpoint of AB. Q is the midpoint of BC and so on. Find the area of the hexagon PQRSTU. Problem 6. An archery target is constructed of five concentric circles such that the area of the inner circle is equal to the area of each of the four rings. If the radius of the outer circle is 12 m, find the width of the band between the second and the third circle. Problem 7. The sum of the distance from a point P to (4,0) and (-4,0) is 9. If the abscissa of P is 1, find its ordinate. Miscellaneous problems on parallelograms Problem 1. The diagonals of a parallelogram measure 16 cm and 24 cm. The shorter side measures 10 cm. (a) Find the area of the parallelogram. (b) Find the measure of the longer side. (c) Find the measure of the smaller angle of the parallelogram. Problem 2. A parallelogram has diagonals 34 in and 20 in and one side measures 15 in. (a) Find the length of the other side. (b) Find the area of the parallelogram. (c) Find the largest interior angle of the parallelogram. Remarkable properties of triangles into which diagonals divide a quadrilateral Problem 1. Let ABCD is a convex quadrilateral and AC and BD are its diagonals intersecting at point P inside the quadrilateral. If quadrilateral is a parallelogram, then triangles APB, BPC, CPD and DPA all have the same area. Problem 2. Let ABCD is an arbitrary convex quadrilateral and AC and BD are its diagonals intersecting at point P inside the quadrilateral. If triangles APB, BPC, CPD and DPA all have the same area, then quadrilateral ABCD is a parallelogram. A trapezoid divided in four triangles by its diagonals Problem 1. ABCD is a trapezoid in which AD:BC = 3:5. If the area of triangle AMD is 315 cm^2, find the area of the trapezoid. A problem on a regular heptagon Problem 1. Given regular heptagon ABCDEFG, a circle can be drawn that is tangent to DC at C and to EF at F. What is radius of the circle if the side length of the heptagon is 1? The area of a regular octagon Problem 1. We cut a regular octagon ABCDEFGH out of a piece of cardboard. If AB = 1 unit, what is the area of the octagon? The fraction of the area of a regular octagon Problem 1. Vertices of a regular octagon are connected as shown in the attached Figure. What fraction of the octagon area is shaded? Try to solve these nice Geometry problems ! Problem 1. Two sides of triangle ABC are AB = 34 cm and AC = 25 cm and their included angle measures 62°. Find the distance of the orthocenter to side AB. Problem 2. The circle circumscribes an equilateral triangle. The area of the circle is 12pi. What is the triangle's perimeter ? Problem 3. The center of a circle is at (-3,-2). If a chord of length 4 is bisected at (3,1), find the length of the radius. Find the angle between sides of folded triangle Problem 1. An isosceles triangle ABC, in which AB = BC = is folded along the altitude BD, so that planes ABD and BDC form a right dihedral angle. Find the angle between side AB and its new position. A problem on three spheres Problem 1. The center of each of three spheres of radius R lies in the surfaces of the other two. Pass a plane containing the centers of the spheres. Find the area common to the three great circles cut from the spheres by this plane. A sphere placed in an inverted cone Problem 1. A sphere is placed in an inverted hollow conical vessel of base radius 5 cm and vertical height 12 cm. If the highest point of the sphere is at the level of the base of the cone, find the radius of the sphere. An upper level Geometry problem on special (15°,30°,135°)-triangle Problem 1. ABC is a triangle with ∠CAB=15° and ∠ABC=30°. If M is the midpoint of AB, find ∠ACM. A great Math Olympiad level Geometry problem Problem 1. In triangle ABC, point X is on side BC such that AX = 13, BX = 13, CX = 5, and the circumcircles of triangles ABX and ACX have the same radius. Find the area of triangle ABC. Nice geometry problem of a Math Olympiad level Problem 1. The area of △ABC is 40. Points P, Q and R lie on sides AB, BC and CA respectively. If AP = 3 and PB = 5, and the area of △ABQ is equal to the area of PBQR, determine the area of △AQC. To navigate over all topics/lessons of the Online Geometry Textbook use this file/link GEOMETRY - YOUR ONLINE TEXTBOOK. This lesson has been accessed 662 times. |
