Plane and solid geometry . prove area of ARCD, = b - h. Argument 1. Draw the rectangle EBCF, having b asbase and h as rt. A DCF and ARE, DC = AR. Also CF = A DCF = A ARE. 6. Now figure ARCF = figuve ARCF,.•. area of ARCD = area of ERCF. But area of ERCF =b -Ji,,\ area of ARCD = b - h. Reasons 1. §223. 2. §232. 3. §232. 4. §211. 5. By iden. 6. § 54, 3. 7. §475. 8. §54,1. 482. Cor. I. ParallelograDis having equal bases and,equal altitudes are equivalent. 483. Cor. n. Any two parallelograms are to each otheras the products of their bases and their alt
Plane and solid geometry . prove area of ARCD, = b - h. Argument 1. Draw the rectangle EBCF, having b asbase and h as rt. A DCF and ARE, DC = AR. Also CF = A DCF = A ARE. 6. Now figure ARCF = figuve ARCF,.•. area of ARCD = area of ERCF. But area of ERCF =b -Ji,,\ area of ARCD = b - h. Reasons 1. §223. 2. §232. 3. §232. 4. §211. 5. By iden. 6. § 54, 3. 7. §475. 8. §54,1. 482. Cor. I. ParallelograDis having equal bases and,equal altitudes are equivalent. 483. Cor. n. Any two parallelograms are to each otheras the products of their bases and their altitudes. Hint. Give a proof similar to that of § 479. 484. Cor. III. (a) Two parallelograms having equalbases are to each other as tJieir altitudes, and (b) two par-allelograms having equal altitudes are to each oilier astheir bases, (Hint. Give a proof similar to that of § 480.) 218 PLANE GEOMETRY Proposition III. Theorem 485. Tlie area of a triangle equals oive lialfthe productof its base and its altitude. T B. A b C Given A ABC, with base b and altitude prove area of A ABC = ^b*h. Argument 1. Through A draw a line 11 CB, and through B draw a line 11 CA, Letthese lines intersect at X 2. Then AXBC is a O. 3. .-. A ABC = 1 O AXBC. 4. But area of AXBC =bh. 5. /. area oiAABC = ^b h. 486. Cor. I. Triangles having equal bases and equ^aZaltitudes are equivalent. 487. Cor. II. Any two tinangles are to each other as theproducts of their bases and tlieir altitudes. Outline of Proof Denote the two A by T and ^Vtheir bases by h and 5, andtheir altitudes by h and li\ respectively. Then T= \h •h and Reasons1. §179. 2. §220. 3. §236. 4. §481. 5. § 54, 8 a. t = \ V . li\ V . h 488. Cor. HI. (a) Two triangles Jiaving equal bases areto earJi other as their altitudes, and (b) two triangleshaving equal altitudes are to each as tlieir bascG. BOOK IV 219 Outline of Proof 489. Cor. IV. A triangle is equivalent to one half ofa parallelogram having the same base and alti
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