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GEOMETRY I

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The Elements of Geometry

Geometry is a mathematical science concerned with the properties of figures, or shapes. When applied to two-dimensional figures, those lying on a flat surface, this science is called plane geometry. Many of the shapes dealt with in science and engineering are plane figures, and the usual concern is the measurement of their areas and the lengths of their boundaries. Plane geometry provides the mathematical tools to accomplish this. There are three basic building blocks of plane geometry: point, line, and surface.

A point is used to indicate position. It has no size- no width, length, or thickness. The point allows a line to be generated, defined as "the path of a moving point." If the point always moves in the same direction, a straight line will be generated. Strictly speaking, a line extends indefinitely. It has no end in either direction. A line segment is a portion of a line. The word line normally means a line segment. A line (line segment) has length but no width or thickness.

A surface may be generated by a moving line that does not move in its own direction. Surfaces have length and width but no thickness. Surfaces may be curved or flat. A surface generated by a line that always moves in the same direction is a flat surface, or plane. For example:

A point moving in the same direction generates a straight line.

A line moving in the same direction generates a flat surface, or plane.

Plane geometry deals with figures on a plane surface. Solid geometry deals with figures that take up space, or have volume, and will be discussed in another article.

Lines and Angles

If two lines lie in the same plane, or on the same plane surface, they must either be parallel or they must intersect, or cross, each other. Parallel lines are two straight lines that lie in the same plane and, even if extended infinitely, would not intersect. An example of both is shown below.

When two lines intersect, they form two pairs of equal angles, and the point of intersection is the vertex of the angles. A single angle is defined when two intersecting lines terminate at the vertex. The unit of measure of an angle is the degree, defined as 1/360 of a complete revolution. Rotating a line about a point (the vertex) can form an angle, and when this line rotates through one complete revolution, back to its starting point, it has turned through 360 degrees (normally written 360°).

There are four types of angles. Two perpendicular lines form a right angle. This angle corresponds to 1/4 of a complete revolution, and so a right angle is exactly 90°. An acute angle is any angle less than 90°. An obtuse angle is greater than 90°. A straight angle is exactly 180° and is called a straight angle because it forms a straight line.

Polygons

Many of the figures of plane geometry are polygons, defined as "closed figures bound by straight line segments." The requirement of straight line segments excludes circles from this category, although circles are also plane, closed figures. There is no limit as to the number of sides that a polygon may have, but as a practical matter, there are only a few types that are normally dealt with. Polygons are named according to their number of sides.

These are the most common polygons.

Other polygons are:

  • Hexagon - 6 sides
  • Heptagon - 7 sides
  • Octagon - 8 sides
  • Decagon - 10 sides
  • Dodecagon - 12 sides

In the general case, there are no restrictions on the lengths of the sides or on the sizes of the angles. A special category of polygons is that of the regular polygon, defined as a polygon in which all sides are of equal length and all angles are equal:

The angles shown in these figures are called interior angles, and the sum of the interior angles for any closed polygon depends upon the number of sides. There is a relationship between the sum of the interior angles and the number of sides for any polygon:

Sum of interior angles = (n - 2)(180) where n is the number of sides.

This relationship is true of all polygons, not only regular polygons. Thus, for a triangle, which has three sides, the sum of the interior angles is always 180. For a quadrilateral, the sum is always 360, etc. This is an important relationship because the size of an unknown angle in a polygon can be determined if all the other angles are known.

For example, what is angle A in the pentagon shown below?

For a pentagon (5 sides), the sum of the interior angles is:

Sum = (5 - 2)(180) = 540

Therefore: A + 110 + 140 + 70 + 160 = 540

A + 480 = 540

A = 60

The two most common polygons used in science and engineering are the triangle and the quadrilateral, and these will be studied in some detail.

Triangles

A triangle is one of the most important polygons. Not only does it arise frequently in geometry, but it also forms the basis of another mathematical science, Trigonometry, which will be studied later. A triangle has three sides, which may be of any length, so long as they form a closed figure. A general triangle is shown below.

The three angles of a triangle, A, B, and C, always add up to 180, regardless of its shape: A + B + C = 180. There are a number of special triangles in which there is a definite relationship among the lengths of their sides. An equilateral triangle is the name given to a regular triangle that has three equal sides. Since it has three equal sides, each of the interior angles is equal, all being 60.

An equilateral triangle has three equal sides and three equal angles.

An isosceles triangle has two equal sides, and the angles opposite these sides are equal:

A scalene triangle has no equal sides or equal angles. This is the most general type of triangle.

The names equilateral, isosceles, and scalene describe triangles according to relationships between the lengths of their sides. Triangles can also be described according to the size of their interior angles. A right triangle has one interior angle equal to 90.

Isosceles and scalene triangles can also be right triangles, but an equilateral triangle can never be a right triangle. The right triangle is an important special category of triangles, and the science of trigonometry is based on its properties.

An acute triangle is one in which all of the angles are acute angles (less than 90). An equilateral triangle is a special acute triangle, since all three angles are equal. Finally, an obtuse triangle contains one angle that is an obtuse angle (greater than 90). The other two angles must be acute angles.

The points at which the sides of a triangle intersect are called vertices. There are clearly three vertices in every triangle. A line originating at one vertex and perpendicular to the opposite side is called an altitude. Every triangle has three altitudes. In an obtuse triangle, two of the altitudes lie outside the triangle.

Quadrilaterals

The four-sided polygon, or quadrilateral, can assume many different forms. The most common polygon has four sides of unequal length and four unequal interior angles. The sum of the interior angles is always equal to 360.

Although this is the most general form of a quadrilateral, it is not particularly useful for engineering applications. There are a number of special quadrilaterals that are named according to the relationships between their sides and interior angles and commonly arise in engineering.

It also follows from the definition of a square that pairs of opposite sides must be parallel.

As the case of the square, the angles are each 90° so opposite sides are parallel. However, one pair of opposite sides may be of different length from the other pair of opposite sides. Notice that a square is a special type of rectangle. All squares are rectangles, but not all rectangles are squares.

Notice that this definition says nothing about the interior angles. If the interior angles were each 90°, then the parallelogram would be a rectangle. Therefore, every rectangle is a parallelogram, but not every parallelogram is a rectangle.

Again, there are no restrictions on the interior angles (except, of course, that their sum must be 360). If the interior angles were each equal to 90°, then the rhombus would be a square.

If the length of side "a" were equal to side "b", then the trapezoid would be a parallelogram.

These five quadrilaterals are the most common types. The next section deals with how to measure the perimeters and areas of triangles and quadrilaterals.

  1. A square is a very special quadrilateral that has four equal sides and four equal angles. Since the angles are all equal, it follows that they are each 90°, or right angles.
  2. A rectangle is a quadrilateral that has four equal angles and has opposite sides of equal length.
  3. A parallelogram is a quadrilateral that has opposite sides equal in length and parallel.
  4. A rhombus is a parallelogram that has four equal sides.
  5. A trapezoid is a quadrilateral that has one pair of parallel opposite sides.

Measurement of Polygons

In dealing with polygons, there are two properties that usually are of concern: the length around the boundary, or perimeter, and the area. The perimeter is easy to measure, since the lengths of the sides are simply added. For example, what is the perimeter of the trapezoid shown below?

The measurement of area is a more complicated problem. The area of a polygon is the number of square units that will exactly cover the polygon, or fit within its perimeter. Picture a square in which each side has a length of 1 unit. That unit may be an inch, a foot, a mile, or any other unit or length. The area of that square is 1 square unit. The area of a polygon is the number of these square units that exactly fit within its boundary, covering the entire polygon. This number need not be a whole number, since fractions of the square unit might be needed to exactly fill the polygon. The areas of triangles and quadrilaterals will be computed in this section.

The area of a square can be found by dividing the square into the proper number of square units. Suppose there is a square in which the length of each side is 3 units. Covering the square with a smaller square in which the length of each side is 1 unit requires 9 of these small squares:

The area of the square is then 9 square units. It would certainly be tedious to have to divide every square into a number of smaller squares and add them up. The same result occurs when multiplying the length of one side of the square by the length of the adjacent side. Since all sides of a square have the same length, the area of a square equals the square of the length of one side:

The area of a rectangle is computed in the same way. In this case, the length of one side is not equal to the length of an adjacent side.

Multiplying the length and the width is equivalent to moving a line of length W through a distance L, which generates the rectangular surface.

The results for the area of a rectangle can be used to find the area of a parallelogram. Remember that the rectangle is a special case of a parallelogram that has all interior angles equal to 90. Shown below is a parallelogram in which a vertical line forms one vertex perpendicular to the base. This line has length h.

Suppose that the figure is cut along the dotted line, and then the triangle piece is moved from the left of the figure and fitted at the right.

This is now a rectangle with length b and width h, so the area is just A = bh. The length h is called the altitude, or height. The altitude is the perpendicular distance between two opposite sides. The length b is called the base of the rectangle.

 

A = bh P = 2b + 2h
P = 2(b + h)

The area of a parallelogram can be used to find the area of a triangle. Shown below is a triangle with sides a, b, and c. From the vertex of the triangle labeled 1, construct a line of

length b parallel to the side of the triangle also having length b. Similarly, from the vertex, labeled 2, construct a line of length a parallel to the side of the triangle also having length a.

A parallelogram has now been formed. Constructing an altitude of length h, the area of the parallelogram will be A = bh. The area of the parallelogram is just twice the area of the original triangle, so the area of the triangle is half the area of the parallelogram.

 

A = 1/2 bh
P = a + b + c

The altitude h can be constructed from any vertex of the triangle, so long as it is perpendicular to the opposite side. In the equation for the area, the side b is the side that is perpendicular to the altitude. This side usually is called the base of the triangle.

The results for the area of a rectangle and the area of a triangle can be combined to find the area of a trapezoid.

Two altitudes of equal length are constructed that divide the trapezoid into two triangles and a rectangle:

Let the base of the left triangle have length x and the base of the right triangle have length y. The area of the left triangle is 1/2 hx, and the area of the right triangle is 1/2 hy. The area of the rectangle is bh, so the total area of the trapezoid is the sum of the area of the three pieces:

 

A = 1/2 hx + 1/2 hy + bh
A = 1/2 h (x + y) + bh
 

But x + y is related to the base of the trapezoid, since x + b + y = B, the length of the lower base. Therefore, x + y = B-b, and so the area of the trapezoid is just:

 

 

A = 1/2 h (B - b) + bh
= 1/2 hB - 1/2 hb + hb
= 1/2 hB + 1/2 hb
A = 1/2 h (B + b) i.e., one-half the product of the altitude and the sum of the bases.

Right Triangles

Three types of triangles have been identified: equilateral, isosceles, and scalene. There is one other very important category of triangles, namely the right triangle. A right triangle contains one 90, or right, angle:

The side opposite the right angle is given the special name hypotenuse; the other sides are called legs. Since the right triangle contains a 90 angle, the other two angles, A and B, must add up to 90 (recall that there is a total of 180 in the angles of every triangle). There is an important rule that relates the lengths of the legs of a right triangle and the hypotenuse. This rule, called the Pythagorean Rule (after the Greek mathematician Pythagoras) states that in every right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.

It should be emphasized that this rule applies only to right triangles. It provides a powerful tool for calculating the length of an unknown side of a triangle when the other two sides are known, even though nothing is known about the interior angles other than that one of them is a right angle. For example, find the length of the leg a in the right triangle shown below.