D AY 159 R OTATION OF 2 D - FUSED FACES I NTRODUCTION In life, we - PowerPoint PPT Presentation

D AY 159 – R OTATION OF 2 D - FUSED FACES

I NTRODUCTION In life, we do encounter figures that are not composed of one geometric figures only. For instance, some design of windows has a rectangle with a semi-circle fused at the top. Some are rectangular with some triangle fused at the top. Our interest is to know the figure they will produce when they sweep through some space as they are opened. In this lesson, we will identify three- dimensional objects generated by rotations of 2D fused faces.

V OCABULARY 2D-fused face This is a figure in a plane that is formed by a combination of two or more distinct plane figures.

Consider the figure below. A F D B C The upper part is a triangle and lower part is rectangle. The two plane figures are said to be fused.

The figure is rotated about an axis lying along line AE. A F D B C E The rectangular part will generate a cylinder while the triangular part will generate a cone. The rotation therefore generates a cylinder with a cone at the top.

Consider a figure below showing an isosceles trapezoid and a square. When it is rotated about an axis coplanar to its line of symmetry, the square generates a cylinder and the trapezoid generates a frustum.

Example 1 Which three dimensional objects are generated when the figure below is rotated about line MN. M N L O P K

Solution The figure can be divided into two right trapezoids and a rectangle as shown. M N L O K P When rotated about MN the trapezoids generate frustums while the rectangle generate a cylinder. Thus the objects generated are two frustums and a cylinder.

HOMEWORK A window is in form of the shape shown below with metal rod at the center about where it rotates to open and close. The window is opened and closed by rotating it 180°. List all three – dimensional objects generated by this rotation.

A NSWERS TO HOMEWORK Hemisphere Cylinder Cone

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