Tower cranes are a common fixture at any major construction site. Trolley The 3D graphic on the left shows a tower crane which contains a long horizontal jib (or working arm), which is the portion of the crane that carries the load. A trolley runs along the jib to move the load in and out. A The drawing on the right shows an elevation of the crane and skip which is to be attached to the trolley. The elevation of the trolley is incomplete. (a) Complete the elevation of the trolley, showing all construction lines. (b) The trolley and skip move along the jib until they reach point A. Show the trolley and skip in the new position after this horizontal movement. (c) When the trolley reaches the end of the jib the skip must be lowered to the ground to be refilled. Translate the skip vertically until the base of the skip reaches the ground. Skip The photo shows a wood screw and a piece of wood. Two pieces of this wood are to be jointed together using the screw. The diagram over (left) shows the Key Principles screw in its initial position A. (a) The screw is to move along the surface of the wood to position A . • Under a horizontal translation all points move the _______ A1 1 Show the screw in the translated position. distance in a ____________ direction A (b) Having positioned the screw, it is to be inserted into the wood vertically until the top of the screw is flush (level) with the top surface of the • Under a vertical translation all points move the ________ wood. Show the screw in the new translated position. distance in a ____________ direction • The shape and size of the object are _________________ DESIGN & COMMUNICATION GRAPHICS Transformation Geometry 1 NAME: ______________________________ DATE: _____________ Pier B B The photograph shows a leisure boat. The diagram over shows the boat docked at pier A. (a) Complete the construction of the boat. X (b) The boat travels from Pier A S64°W for a distance of 105mm. Show the translation of the boat into this po- sition at sea. (c) The boat then travels to Pier B where point X on the Pier A boat docks at point B. Show the translation of the boat into this position. The photograph shows the door number for a house. The diagram shows this number 4 setup using a square grid. (a) The figure is subject to a translation. Find the image of the door number when point P moves to P1 under the translation. (b) The image is translated N110°E through a distance of 100mm. Construct the image figure under this translation. P1 Key Principles • Under a translation, all points on an object are moved the _______ distance in the _______ direction • Corresponding lines on the object and its image are always ___________ • A line must be ________________ to locate its midpoint P • A line that bisects another line at right angles is called a _________________ bisector DESIGN & COMMUNICATION GRAPHICS Transformation Geometry 2 NAME: ______________________________ DATE: _____________ The photograph shows a training cone in contact with a football. The diagram below shows the elevation and plan of the football S and an eleva- tion of the cone C. The centre of the sphere and cone are located on a line par- allel to the vertical plane (XY line) in plan. (a) Translate the elevation of the cone until it is in contact with the sphere. (b) Project a plan of the solids in contact. (c) Show the projections of the point of contact between the solids. Cone C The cone is translated horizontally along the Horizontal Plane. The extreme generator of the cone will be tangential to the sphere. Key Principles • Under a translation, a line is mapped onto a __________ line • The extreme generator of the cone will be a ________________ to the circle (sphere) in elevation • The elevation of the point of contact is located by drawing a line from the centre of the sphere _________________ to the tangent DESIGN & COMMUNICATION GRAPHICS Transformation Geometry 3 NAME: ______________________________ DATE: _____________ The photograph shows a road sign which is commonly seen at roundabouts. The arrows indicate to drivers the direction to follow. These arrows are translated across the rectangular sign. The diagram on the right shows a portion of the rectangular outline and the initial position of the arrow. (a) Complete the road sign by translating the arrow. (b) Shade the road sign appropriately. Arrows are equally spaced School The diagram on the right shows the plan of school and the position of a bus stop on an adjacent road. Determine the position of the centerline of the path and the zebra crossing so that the path from the main entrance S to the bus-stop B is of minimum distance for students walking to the school. S B Bus Stop B Key Principles • Parallel lines always remain __________________ • Under a horizontal translation all points move the _______ distance in a ____________ direction DESIGN & COMMUNICATION GRAPHICS • The shortest distance between two points is a ____________ ____________ Transformation Geometry 4 NAME: ______________________________ DATE: _____________ The photograph shows the Toastrack Hotel in Manchester. The building is The photograph shows the inside of a building in Barcelona. The roof structure is generated by translating the parabola ABC in a vertical position along the generated by translating the parabola ABC in a vertical position along the straight horizontal line BD. Also shown is the incomplete elevation and plan of line BD. Also shown is the elevation and incomplete plan of the roof. the structure. Complete the plan of the roof showing all construction lines. Complete the elevation and plan of the hotel D B B B B D A,C A C x y A,C A C x y A A B B C C As the parabola translates along the inclined line BD, the points A and C on the parabola move below the HP. The intersection of the HP and translated surface creates the outline of the plan. DESIGN & COMMUNICATION GRAPHICS Transformation Geometry 5 Parabola ABC is in the vertical position with the The parabola translates (pt B) along the line BD line BD horizontal. to form the hotel shape. NAME: ______________________________ DATE: _____________ Identify all axes of symmetry in each of the following objects: Rotated Reflection in nature Taj Mahal Square Star of David Axis of symmetry Isosceles Triangle Equilateral Triangle Mercedes Logo Maple Leaf Stop Sign Key Principles • Axial symmetry is symmetry about a ______________ • All symmetrical objects have at least ________ axis of symmetry • An object can have more than __________ axis of symmetry. DESIGN & COMMUNICATION GRAPHICS Transformation Geometry 6 Alloy Wheel Aeroplane Wing NAME: ______________________________ DATE: _____________ Complete each of the designs below using axial symmetry. Complete the drawing of the butterfly. Y A quilt pattern A football jersey Pizza Broom X Complete the graph of the parabola Key Principles • An object and its ______ are the same shape and ______ • A point and its image are the same distance from the _______ of reflection • The axis of reflection is the perpendicular bisector of a line joining a point and its _______ DESIGN & COMMUNICATION GRAPHICS Transformation Geometry 7 NAME: ______________________________ DATE: _____________ Plot the locus of the white ball if it is to strike the yellow ball so that it will enter the corner pocket marked C. P and Q represent two houses which require connection to a gas main represented by the line G. Find the location of a common connection point on the main supply line which will give the shortest length of pipe required. P Q C G Show how the point of contact between the ellipse and the tangent is obtained. Key Principles • An object and its are the same shape and ______ • A point and its image are the same distance from the _______ F1 F2 of reflection • The axis of reflection is the perpendicular bisector of a line join- ing a point and its _______ Law of reflection of light DESIGN & COMMUNICATION GRAPHICS Transformation Geometry 8 NAME: ______________________________ DATE: _____________ When a ball is hit, the path the ball takes is in the shape of a parabola. D The construction of the parabola begins with a fixed point for a focus and a line for a directrix. The drawing below shows the directrix DD of a parabola with the focal point at F. Draw the parabola showing all construction lines. D D F F Q D The drawing shows the directrix DD and the focus F of a parabola. A series of points (such as Q) are marked on the directrix. The axis of reflection which maps Q onto F has been determined as shown. This axis of reflection is a tangent to the parabola. Locate the axes of reflection for the points on the directrix below the axis to generate a view of the skateboard ramp shown on the left. Key Principles • The set of points equidistant from the focus and the ____________ is the parabola DESIGN & COMMUNICATION GRAPHICS Transformation Geometry 9 • Every point on the perpendicular bisector of a line segment is equidistant from the ___________ of the line NAME: ______________________________ DATE: _____________ The photograph shows a clock which is hexagonal in shape. The diagram below The photograph shows a football in which all the black areas are shows an equilateral triangle ABC. pentagonal in shape. The diagram below shows an isosceles triangle LMN. The side MN forms one side of the pentagon. (a) Rotate the triangle about point B to create the remainder of the clock.