# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/४११

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215
CHAPTER VII--MEASUREMENT OF AREAS.

reference is further constructed) with the aid of the measurements of the base and the perpendicular-side (of the immediately derived quadrilateral, above referred to, used as bījas. Then, with the aid of these two last derived secondary quadrilaterals, all the required) quantities appertaining to the quadrilateral with three equal sides are (to be obtained) as in the case of the quadrilateral with two equal sides.

An example in illustration thereof.

102${\displaystyle {\tfrac {1}{2}}}$. In relation to a quadrilateral with three equal sides and having 2 and 3 as its bījas, give out the measures of the top-side, of the base, of (any one of) the (equal) sides, of the perpendicular (from the top to the base), of the diagonal, of the (lesser) segment (of the base) and of the area.

The rule for arriving at the measures of the top-side, of the base, of the (lateral) sides, of the perpendiculars (from the ends of the top-side to the base), of the diagonals, of the segments (of the base) and of the area, in relation to a quadrilateral the sides of which are (all) unequal :-

103${\displaystyle {\tfrac {1}{2}}}$. [1] With the longer and the shorter diagonals (of the two derived rectangular quadrilateral figures related to the two sets

103${\displaystyle {\tfrac {1}{2}}}$.^  The rule will be clear from the following algebraical representation. Let a, b, and c, d, be two sets of given bījas. Then the various required elements are as follow:-

Lateral sides ${\displaystyle =2ab(c^{2}+d^{2})(a^{2}+b^{2}){\text{ and }}(a^{2}-b^{2})(c^{2}-d^{2})}$${\displaystyle (a^{2}+b^{2})}$
Base ${\displaystyle =2cd(a^{2}+b^{2})(a^{2}+b^{2}).}$
Top-side ${\displaystyle =(c^{2}-d^{2})(a^{2}+b^{2})(a^{2}+b^{2}).}$
Diagonals ${\displaystyle ={\Bigg \{}(a^{2}-b^{2})\times 2cd+(c^{2}-d^{2})2ab{\Bigg \}}\times (a^{2}+b^{2}){\text{; and }}}$
${\displaystyle {\Bigg \{}(a^{2}-b^{2})(c^{2}-d^{2})+4abcd{\Bigg \}}\times (a^{2}+b^{2})}$
Perpendiculars ${\displaystyle ={\Bigg \{}(a^{2}-b^{2})\times 2cd+(c^{2}-d^{2})2ab{\Bigg \}}\times 2ab{\text{; and }}}$
${\displaystyle {\Bigg \{}(a^{2}-b^{2})(c^{2}-d^{2})+4abcd{\Bigg \}}\times (a^{2}-b^{2})}$
Segments ${\displaystyle ={\Bigg \{}(a^{2}-b^{2})\times 2cd+(c^{2}-d^{2})\times 2ab{\Bigg \}}(a^{2}-b^{2}){\text{; and }}}$
${\displaystyle {\Bigg \{}(a^{2}-b^{2})(c^{2}-d^{2})+4abcd{\Bigg \}}\times 2ab}$