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## GEOMETRY IMenu
## The Elements of GeometryGeometry 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 A A 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. ## Lines and AnglesIf 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 There are four types of angles. Two perpendicular lines form a ## PolygonsMany of the figures of plane geometry are 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 The angles shown in these figures are called Sum of interior angles = ( 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 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. ## TrianglesA 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 An An A The names 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 The points at which the sides of a triangle intersect are called ## QuadrilateralsThe four-sided polygon, or 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. - 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. - A
**rectangle**is a quadrilateral that has four equal angles and has opposite sides of equal length. - A
**parallelogram**is a quadrilateral that has opposite sides equal in length and parallel. - A
**rhombus**is a parallelogram that has four equal sides. - A
**trapezoid**is a quadrilateral that has one pair of parallel opposite sides.
## Measurement of PolygonsIn 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 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 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
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 length A parallelogram has now been formed. Constructing an altitude of length
The altitude 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
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:
## Right TrianglesThree 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 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 |