THE TRAPEZOID 370. Theorem. The area of a trapezoid is equal to half the product of its altitude and the sum of its bases. A =1⁄2h(b+b′). Given the trapezoid DEFH with altitude h and bases b and b'. To prove area of trapezoid DEFH=1⁄2h(b+b′). Proof. Draw the diagonal EH. Area of AEFH = b'h. Area of ADEH = bh. ... area of trapezoid DEFH=h(b+b'). $362 Why? Why? 371. Theorem. The area of a trapezoid equals the product of its altitude and its median. Show that the median = (b+b'). EXERCISES COMPUTATIONS 1. Find the area of a triangle having a base of 4 ft. 7 in. and an altitude of 8 ft. 6 in. Of one having a base of 5.4 ft. and an altitude of 9 in. 2. Find the altitude of a triangle having an area of 160 sq. ft. and a base of 20 ft. Of one having an area of 27.3 sq. in. and a base of 8.1 in. 3. In a trapezoid, A stands for the area, b and b' respectively for the bases, and h for the altitude. (1) Given b = 10 in., b' 7 in., h=8 in.; find A. (2) Given b=33 in., b′ =67 in., h=53 in.; find A. (4) Given b=18 ft., h = 10 ft., A = 150 sq. ft.; find b'. 4. The sides of a triangle are 7 in., 10 in., and 14 in. respectively. Is its shape fixed? Is its area fixed or may it be changed? 5. The four sides of a quadrilateral are 29 in., 31 in., 42 in., and 37 in., respectively. Can its area be found? Why? 6. The two bases of a trapezoid are 14 and 10 respectively and the other two sides are 6 and 5. Does it have a definite area? 7. Find the area of a board that is 14 ft. long and has parallel ends respectively 18 in. and 12 in. 8. Find the number of square feet in the floor of the room shown in Fig. 1. 9. Find the area of the cross section of the channel iron shown in Fig. 2. 10. Write a formula for finding the area of the cross section of a channel iron, using the letters as given in the figure. Ans. A td+b(s+n). 11. Using the formula derived in the preceding Exercise, find the areas of the cross sections of channel irons of the following dimensions: Ans. 6.02 sq. in. 12. Write a formula for finding the area of the cross section of an I-beam, using the letters as given in the figure. Ans. A dt +2b(n+s). 13. Using the formula derived in the preceding Exercise, find the areas of the cross sections of I-beams of the following dimensions: 14. The base of a triangle is 3 in. longer than the altitude, and the area is 44 sq. in. Find the lengths of the base and altitude. 15. The altitude of a triangle is 3 times the base, and the area is 37 sq. in. Find the lengths of the base and altitude. Ans. 5 in. and 15 in. 16. The difference between the base and the altitude of a triangle is 6 in., and their sum is 36 in. Find the area of the triangle. Ans. 157 sq. in. If the 17. The altitude of a triangle is 7 in. longer than the base. altitude is decreased by 4 in. and the base increased by 6 in., the area is increased by 25 sq. in. Find the altitude and base. *18. The figure is a trapezoid with dimensions as given. Find the distance from the longer base DE to the line AB which is drawn parallel to DE and G divides the area into two equal parts. D 1500' 190 E CONSTRUCTIONS 1. Construct a triangle equivalent to a given triangle and having one of its sides equal to a given line. 2. Construct a triangle equivalent to a given triangle and having one of its angles equal to a given angle. 3. Divide a triangle into four equivalent triangles by lines through one of its vertices. 4. Construct a triangle equivalent to a given square. 5. Construct a triangle equivalent to a given triangle AHK and having its base double the length AH. 6. Construct a triangle equivalent to of a given triangle. 7. On the base of a given triangle, construct a right triangle equal in area to the given triangle. 8. On the base of a given triangle, construct an isosceles triangle equal in area to the given triangle. 9. On the base of a given triangle, construct a rectangle equal in area to the triangle. PRACTICAL METHODS 372. Practical Methods for Finding the Area of a Polygon. The area of any polygon may be found by dividing it into triangles and finding the sum of the areas of the triangles. B G E H K D L C The area of a polygon may also be found by drawing the longest diagonal and drawing perpendiculars to this diagonal from the vertices. This will divide the polygon into triangles and trapezoids or rectangles, of which the areas can be found. The sum of these areas is the area of the polygon. 373. Trapezoidal Rule. An area between a curved line and a straight line is sometimes found approximately as follows: (1) Divide the straight line AB, called the base line, into a number of equal parts of length d. (2) Measure the lines, called offsets, drawn perpendicular to AB from the points of division to the curved line, their lengths being y1, y2, y3, etc. (3) Consider each part, as ACDF, a trapezoid and find the sum of their areas, which is the area required, approximately. Show that this gives the following rule which is known as the trapezoidal rule: To half the sum of the first and last offsets add the sum of all the other offsets, and multiply this sum by the common distance between the offsets. By drawing a straight line, as AB in the figure, through a surface bounded by an irregular line, the area of each part can be computed approximately by the trapezoidal rule. 374. Areas on Square-ruled Paper. A very useful practical way to estimate B the area of any plane surface is to draw it on square-ruled paper The figure will usually be drawn to some scale that uses a side of one of the squares as a unit of length. The squares enclosed in the figure can then be counted, and thus the area of the surface may be estimated. A convenient way is to count as whole squares those which are entirely within or more than half within the figure, to disregard those squares which are more than half without the figure, and to count every other one of the squares that are half within and half without. EXERCISES 1. Draw a rectangle 2 in. by 3 in. with a semicircle on one of the long sides. Measure 6 offsets from the other long side, and compute the area 4. Estimate the area of this figure if it is drawn on a scale of 1 in. to a side of a square. 5. Take leaves of various trees and find their areas by drawing their outlines on square-ruled paper. 6. Estimate the area of this figure if it is drawn on a scale of 1 ft. to a side of a square. If the area of this figure is 150.8 sq. ft., what is the per cent of error in your estimate? *7. By the trapezoidal rule determine the area of a lake, using a map of the lake. It is necessary to know the scale to which the map is drawn. Draw the outline of the lake on square-ruled paper and determine its area. *8. To determine the amount of flow of water in a river that is 120 ft. wide, soundings were made for each 10 ft. of width. Thus 13 measurements were made as shown in the figure. Find the area of the cross section in square feet. 16 5.1 1 -120 5.5 5.?! 5.8 5.7 5.4' 4.71 5.1' |