Page:Indian mathematics, Kaye (1915).djvu/76

 44. Tell the perpendicular drawn from the intersection of strings mutually stretched from the roots to summits of two bamboos fifteen and ten hastas high standing upon ground of unknown extent? Answer—6. 45. Of a quadrilateral figure whose base is the square of four and the face two hastas and altitude twelve, the flanks thirteen and fifteen, what is the area? Answer—108. 46. In the figure of the form of a young moon the middle length is sixteen and the middle breadth is three hastas. By splitting it up into two triangles tell me, quickly, its area? Answer—24. 47. The sides of a quadrilateral with unequal sides are $\scriptstyle{13\times 15}$, $$\scriptstyle{13\times 20}$$ and the top side is the cube of 5 and the bottom side is 300. What are all the values here beginning with that of the diagonals? Answers—315, 280, 48, 252, 132, 168, 224, 189, 44100. If $$\scriptstyle{A^2+B^2=C^2}$$, and $$\scriptstyle{a^2+b^2=c^2}$$ then the quadrilateral Ac, Bc, aC, bC is cyclic and the diagonals are $$\scriptstyle{Ab+aB}$$ and $\scriptstyle{Aa+Bb}$, the area is $\scriptstyle{\frac{1}{2}(ABc^2+abC^2)}$, &amp;c. In the present case $\scriptstyle{A=15}$, $\scriptstyle{B=20}$, $\scriptstyle{C=25}$; $\scriptstyle{a=5}$, $\scriptstyle{b=12}$, $\scriptstyle{c=13}$. The diagonals are 315, 280; the area 44100. For full details see the Līlāvatī, § 193. 48. O friend, who knowest the secret of calculation, construct a derived figure with the aid of 3 and 5 as elements, and then think out and mention quickly the numbers measuring the perpendicular side, the other side and the hypotenuse? Answer—16, 30, 34. That is construct a triangle of the form $\scriptstyle{2mn}$, $\scriptstyle{m^2-n^2}$, $\scriptstyle{m^2+n^2}$, where m=5, n= 3. L=the Līlāvatī, V=Vīja Gaņita, both by Bhāskara, M=Mahāvīra, S=SrīdharaŚrīdhara [sic], C=Chaturveda.