A complete and practical solution book for the common school teacher . FIG. 39. MENSURATION. 165 (7) Whole area grazed over is sq. ft.+ sq. sq. ft. PROBLEM 80 cents a rod, what would be the cost of fencing- a field in theform of an equilateral triangle, its altitude being 6 Solution. (1) Let ABC be the triangle. (2) AO bisects the angle A, then angle OAD=30°. Then AODis a right-angled triangle. (3) Now, OD+DE = CO, or the ra- dius of the circle. (4) CO or AO=| of 6 rd. = 4 rd. (5) AB = AOV3, or 4V3= rd. (6) The perimeter is rd. (7
A complete and practical solution book for the common school teacher . FIG. 39. MENSURATION. 165 (7) Whole area grazed over is sq. ft.+ sq. sq. ft. PROBLEM 80 cents a rod, what would be the cost of fencing- a field in theform of an equilateral triangle, its altitude being 6 Solution. (1) Let ABC be the triangle. (2) AO bisects the angle A, then angle OAD=30°. Then AODis a right-angled triangle. (3) Now, OD+DE = CO, or the ra- dius of the circle. (4) CO or AO=| of 6 rd. = 4 rd. (5) AB = AOV3, or 4V3= rd. (6) The perimeter is rd. (7) X.$80 = $, cost of fencing the PIG. 40. NoTK.—As AOD is a right triangle and the angle OAD=30°, thenAO=4 and OD=2 rd. It follows that the hjpothenuse is twice the sideopposite the acute angle of a right triangle which is 30°. Also if thehjpothenuse be given, then the side opposite the 30° angle is half ofthe hypothenuse. Prove that AOV3=AB. A figure with six sides is called ahexagon; the side is equal to the radiusof the circumscribing circle. Now, ifI unite the alternate angles of the reg-ular hexagon, as AB, BC, and CA, Ihave a regular triangle, called anequilateral triangle. Join AD, thenADB is a right triangle. Then OD= _ AB8=ADa—DB8=4AOa—AO», or(DB8)=3AO*. Whence AB=AOV3;that is,the side of an equilateral tri-angle is equal to the radius of the cir-cumscribing circle multipied by the PIG. root of 3. PROBLEM 350. The area of an equilateral whose base falls on the diameter andwhose vertex falls on the middle point of the arc of a semicircle, is 80sq. ft.: what is the diameter of the semic
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