The length of AB is called the base (b), and thelength of DC, the altitude (h); therefore, your for-mula for determining the area of an obliquetriangle is again A = 1/2bh.You must remember that in a right triangleh corresponds to the length of one of the sides,while in an oblique triangle it does not. Therefore,for a right triangle with the length of the sidesgiven, you can determine the area by the formulaA = 1/2bh. For an oblique triangle with thelength of the sides given, you cannot use thisformula unless you can determine the value of h,Later in this chapter you will learn trigonometricmethods of determining areas of various formsof triangles on the basis of the length of the sidesalone.Area of a Rhombus or RhomboidFigure 1-9 shows a rhomboid, ABCD. If youdrop a perpendicular, CF, from L C to AD, andproject another from LA to BC, you will createtwo right triangles, AAEB and ACFD, and therectangle AECF. It can be shown geometricallythat the right triangles are similar and equal.You can see that the area of the rectangleAECF equals the product of AF x FC. The areaof the triangle CFD equals 1 /2(FD)(FC). Becausethe triangle AEB is equal and similar to CFD, thearea of that triangle also equals 1/2(FD)(FC).Therefore, the total area of both triangles equals(FD)(FC). The total area of the rhomboid equalsthe area of the rectangle AECF + the total areaof both triangles.The total area of the rhomboid equals(AF)(FC) + (FD)(FC), or (AF + FD)(FC). ButAF + FD equals AD, the base. FC equals thealtitude. Therefore, the formula for the area ofa rhomboid is A = bh. Here again you mustFigure 1-10.-Trapezoid.remember that h in a rectangle corresponds to thelength of one of the sides, but h in a rhombus orrhomboid does not.Area of a TrapezoidFigure 1-10 shows a trapezoid, ABCD. If youdrop perpendiculars BE and CF from points Band C, respectively, you create the right trianglesAEB and DFC and the rectangle EBCF betweenthem. The area of the trapezoid obviously equalsthe sum of the areas of these figures.The area of AAEB equals 1/2(AE)(FC),the area of ADFC equals 1/2(FD)(FC),and the area of EBCF equals (EF)(FC). There-fore, the area of the trapezoid ABCD equalsl/2(AE)(FC) + (EF)(FC) + 1/2(FD)(FC), orHowever, 2EF = EF + BC. Therefore, the areaof the trapezoid equalsBut AE + FD + EF = AD. Therefore, the areaof the trapezoid equalsAD and BC are the bases of the trapezoid andare usually designated as bl and b2, respectively.FC is the altitude and is generally designated ash. Therefore, the formula for the area of atrapezoid isFigure 1-9.-Rhomboid.1-11
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