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Chapter 9—Basic Geometry

9-1 Introducing Geometric Figures

 When you complete the work for this section, you should be able to: Describe the nature of points, lines, and line segments. Define, identify, and sketch plane figures known as a triangle, square, rectangle, parallelogram, trapezoid, and circle. Define, identify, and sketch solid figures know as rectangular solid, cube, pyramid, cylinder, sphere, and cone.

Geometry is the study of shapes, namely points, lines, angles, surfaces, and solids. Why should we study geometry? There are a lot of practical applications that are mostly concerned with measuring or calculating the size of things everything from measuring our waistlines to figuring out the number of rolls of wallpaper required for fixing up the living room. In addition to endless practical applications, geometry also helps us train our minds to think analytically about all kinds of problems, even problems that do not seem to be directly related to mathematics.

Points and Lines Basic geometry is built upon a foundation of points, lines, and line segments

 Definitions A point is a location in space. A point has no size — no height or width. It is infinitely small. A point is usually indicated with a dot. A line is a collection of points arranged a straight row. A line contains an infinite (unending) number of points. The arrows at the ends of a line indicate the line has no beginning or end. A line segment is a portion of a line that stretches between two points.

Points have no dimensions. They are purely imaginary and invisible. But we need to work with them, because we need them to define lines, and we need lines to represent figures we want to measure and analyze. So we usually indicate a point with a dot.

What is a line?  Technically speaking, a line is made up of a countless number of points that are lined up in a straight row. Since points are imaginary, invisible things, it figures that a line is also an imaginary, invisible thing. We usually indicate a line with a ... well, a line.

Technically speaking once again, a line has no beginning or end. The imaginary, invisible line stretches out to infinity in both directions. Such a thing has no practical application in the real-world, so we draw lines on paper, on a computer screen, or in the sand. And we give them a starting point and and ending point that we can clearly see.  A line that we give a starting point and ending point is called a line segment.

Plane Figures

A plane is an imaginary, invisible flat surface that extends indefinitely in two dimensions.  In order to make any practical use of planes, we must apply some limits. We can use some points, lines, and curves to create those limits. And there are six basic ways to combine points, lines, and curves to make sense of the notion of plane figures: triangle, square, rectangle, parallelogram, trapezoid, and circle. There are far more actual plane figures than anyone would care to count; however, they are all made up from variations and combinations of these six basic plane figures.

 Definitions A plane is a flat surface that is completely enclosed on all sides. A plane figure is a flat figure that is defined by points, line segments, and curves. The six plane figures of basic geometry.

 Definitions A triangle is a plane figure that consists of three sides and three angles. A square is is a plane figure that has four equal sides and four right angles. A rectangle is plane figure that has four sides and four right angles. Parallel sides are of equal length. A parallelogram is a plane figure that has four sides. Parallel sides are of equal length, but not all angles are right angles. A trapezoid is is a plane figure that has four sides. Only one set of sides is parallel. There are no right angles. A circle is the set of points that are all the same distance from a single point at the center of the circle.

What is a curve?

 Technically, a curve is an imaginary, invisible figure composed of  points that are aligned so that each one deviates a bit more from straight-line formation. This kind of curve is also known as an arc.

What is a right angle?

 A right angle is an exact square angle; a full quarter turn; a 90-degree angle. What are parallel lines?

 The distance between parallel lines is the same at every point.The distance between lines that are not parallel is different for every point along their lengths. A triangle is a plane figure that consists of three sides and three angles. All three sides and all three angles may be equal, all three sides and all three angles may be unequal, and there are all sorts of combinations of equal and unequal sides and angles. In other words, there are a lot of different kinds of triangles. Here are some samples of different kinds of triangles. Although they might appear to be different in many respects they have these features in common:

• They are plane figures
• They have three sides
• They have three angles

Squares and rectangles are plane figures that have four sides and four right angles.
• For a square, all four sides must be equal.
• For a rectangle, parallel sides must be equal. Squares and Rectangles.

 Thinking Mathematically A square is just a special type of rectangle. A square is a rectangle that happens to have four equal sides.

parallelogram  is a very close cousin of the rectangle — it is a plane figure with four sides, and the opposite sides are parallel. The big difference is that the angles of a trapezoid are not right angles. Comparing rectangles and parallelograms. A trapezoid is a closed, 4-sided plane figure that has just two parallel sides. This makes the figure quite different from rectangles and parallelograms that have two sets of parallel sides.

Most people recognize a circle when they see one. And they can draw circles, too. But few people can give a good mathematical definition of a circle. Here is a suitable mathematical definition:

A circle is the set of points that are all the same distance from a single point at the center of the circle. The distance from the center of a circle to its outer edge is called the radius.

Solid Figures

You have seen that plane figures are built from points and lines that are arranged in certain ways in a flat, two-dimensional world of length and width. Now you can extend that idea to solid figures. Solid figures are built from points, lines, and plane figures that are arranged in three-dimensional space--a world that has length, width, and depth. Here are the solid versions of the six basic plane figures described in the previous section of this lesson. Six solid figures of basic geometry.

 Definitions A rectangular solid is a 3-dimensional version of a rectangle. It has six rectangular sides. All angles are right angles. Opposite sides are parallel and have the same length. A cube is 3-dimensional version of a square. It has six square sides. All angles are right angles. All sides are parallel and have the same length. A pyramid has five faces. One face (called the base)  is square or rectangular, and the remaining sides are triangular. The point at the top is called the apex. A cylinder is the volume enclosed by a two parallel circles. A sphere is 3-dimensional version of a circle. A cone can be considered a circular pyramid. Whereas the base of a pyramid is a square or rectangle, the base of a cone is a circle.

Note: There are many variations of these six basic solid figures. But if you can master the definitions and ways of handling these figures, you will be adequately prepared to learn about the their many variations.

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