If the sides of a rectangle are 4 and 5 feet respectively, the area is 20 square feet. Thus we have the proportion 20: 180:: the square of 4 : the square of required corresponding side. Therefore the square of this side= 16 × 180 = 16 × 9=144; thus this side = 12 feet; and therefore the other required side 15 feet. = (4) An equilateral triangle and a circle have the same perimeter: compare their areas. Suppose each side of the triangle to be 1 foot; then the area is 43301 square feet. The perimeter of the triangle is 3 feet. If the perimeter of a circle be 3 feet, the area of the circle will be found by Chap. xvI. to be '7162 square feet. Divide 7162 by 43301; the quotient is 165... Thus the area of the circle is 1'65... times the area of the equilateral triangle. We shall obtain the same final result whatever be the length of the side of the equilateral triangle. If, for example, we supposed each side to be 7 feet, we shall obtain for the areas of the triangle and of the circle respectively 49 times the former values, but the proportion of the areas will remain unchanged. EXAMPLES. XIX. 1. A field containing 3600 square yards is laid down on a plan to a scale of 1 inch to 10 feet: find the number of square inches of the plan it will occupy. 2. A field containing 6 acres is laid down on a plan to a scale of 1 inch to 20 feet: find how much paper it will cover. 3. Determine the scale used in the construction of a plan upon which every square inch of surface represents a square yard. 4. Determine the scale used in the construction of a plan upon which a square foot of surface represents an area of ten acres. 5. A field is ten thousand times as large as the plan which has been made of it; find what length on the plan will represent a length of 20 yards in the field. 6. An estate, which has been surveyed, is one hundred million times as large as the plan which has been made of it: express the scale of the plan in terms of inches to a mile. 7. The sides of a rectangle are in the proportion of 2 to 3; and the area is 210 square feet: find the sides. 8. The sides of a triangle are in the proportion of the numbers 13, 14, and 15; and the area is 24276 square feet: find the sides in feet. 9. The sides of a triangle are in the proportion of the numbers 7, 15, and 20; and the area is 2226 square feet: find the sides in feet. 10. An equilateral triangle and a square have the same perimeter: compare their areas. 11. A square and a regular hexagon have the same perimeter: compare their areas. 12. A circle and a square have the same perimeter : compare their areas. 13. A circle and a regular hexagon have the same perimeter: compare their areas. 14. Find the side of an equilateral triangle, so that the area may be 100 square feet. 15. Find the side of a regular hexagon, which shall be equal in area to an equilateral triangle, each side of which is 150 feet. 16. Find the radius of a circle, such that the area of a segment corresponding to an angle of 90° may be 50 square feet. 17. One side of a triangle is 15 feet; it is required to divide the triangle into five equal parts by straight lines parallel to one of the other sides: find the distances from the vertex of the points of division of the given side. 18. An equilateral triangle and a square have the same area: compare their perimeters. 19. The parallel sides of a trapezoid are respectively 16 and 20 feet, and the perpendicular distance between them is 5 feet; it is required to divide the trapezoid into two equal trapezoids: find the distance of the dividing straight line from the shorter of the parallel sides. 20. The side of a square is 12 feet; the square is divided into three equal parts by two straight lines parallel to a diagonal: find the perpendicular distance between the parallel straight lines. 21. A circle and a regular polygon of twelve sides have the same perimeter: shew, by Arts. 99 and 167, that the area of the circle is polygon. 3.2154 times the area of the 121 FOURTH SECTION. VOLUMES. XX. DEFINITIONS. 212. WE shall commence this part of the subject with definitions of some terms which we shall have to employ. Although it is convenient to collect the definitions in one Chapter, it is not necessary for the beginner to study them closely all at once; it will be sufficient to read them with attention, and then to recur to them hereafter as occasion may require. 213. Parallel planes are such as do not meet one another, although produced. Thus the floor and the ceiling of a room are parallel planes. 214. A straight line is said to be at right angles to a plane, or perpendicular to a plane, when it makes right angles with every straight line which it meets in that plane. This is the strict geometrical definition; but without regarding it, the student will gain an adequate idea of what is meant by a perpendicular to a plane from considering the illustration afforded by a straight rod when fixed in the ground so as to be upright. So also the student will readily understand when one plane is perpendicular or at right angles to another, without a strict geometrical definition. Thus the walls of a room are perpendicular to the floor and to the ceiling; and a door while moving on its hinges remains perpendicular to the floor and to the ceiling. 215. A parallelepiped is a solid bounded by six parallelograms, of which every opposite two are equal and in parallel planes. The diagram represents a parallelepiped. ABCD and EFGH are equal parallelograms in parallel planes; ABFE and DCGH are equal parallelograms in parallel planes; and ADHE and BCGFare equal parallelograms in parallel planes. A parallelepiped is called rectangular when the six bounding parallelograms are rectangles, and oblique when they are not. A common brick furnishes an example of a rectangular parallelepiped. A rectangular parallelepiped which has its six bounding rectangles all equal is called a cube. This will be found equivalent to saying that a cube is a solid bounded by six equal squares, of which every opposite two are in parallel planes. 216. Plane figures which form the boundaries of solids are called faces of the solid; the straight lines which form the boundaries of the plane figures are called edges of the solid. Thus a parallelepiped has six faces and twelve edges. 217. A prism is a solid bounded by plane rectilineal figures, of which two are equal and in parallel planes, and the rest are parallelograms. The two bounding figures which are equal and in parallel planes are called the ends of the prism. The diagram represents a prism having for its ends the equal pentagons ABCDE and FGHIK; these figures are in parallel planes. The other E K H bounding figures of the solid are parallelograms, as ABGF, BCHG, and so on. Such a prism is called a pentagonal prism; if the ends are hexagons, the prism is called a hexagonal prism; and so on. |