पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/४३०

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GAṆITASĀRASAṄGRAHA.

An example in illustration thereof.

153-153${\displaystyle {\tfrac {1}{2}}}$. In the the case of a quadrilateral figure with unequal sides, the (given) accurate measure of the area is 90. And the product of 5 multiplied by 9, as multiplied by 10, 18, 20 and 36 respectively, gives rise to the (four given) divisors. Tell me quickly, after calculating, the numerical values of the top-side, the base and (other) sides.

The rule for arriving at the numerical value of the sides of an equilateral triangular figure possessing a given accurately measured area, when the value of (that) accurately measured area is known :

154${\displaystyle {\tfrac {1}{2}}}$. Four times the (given) area is squared. (The resulting quantity) is divided by 3. The quotient (so) obtained happens to be the square of the square of the value of the side of an equilateral triangular figure,

An example in illlustration thereof.

155${\displaystyle {\tfrac {1}{2}}}$. In the case of a certain equilateral triangular figure, the given area is only 3. (calculate and tell me the value of (its) side.

After knowing the exact numerical measure of a (given) area, the rule for arriving at the numerical values of the sides, the base and the perpendicular of an isosceles triangular figure having that same accurately measured area (as its own):-

156${\displaystyle {\tfrac {1}{2}}}$. In the case of the isosceles triangle (to be so) constructed, the square root of the sum of the squares of the quotient obtained by dividing the (given) area by an optionally chosen quantity, as also of (that) optionally chosen quantity, gives rise to the value of the side: twice the optionally chosen quantity gives the measure of the base and the area divided by

154${\displaystyle {\tfrac {1}{2}}}$. The rule here given may be seen to be derived from the formula for the area of an equilateral triangle, viz, area ${\displaystyle =a^{2}\times {\sqrt {\tfrac {3}{4}}}}$ where 'a' is the measure of a side,

156${\displaystyle {\tfrac {1}{2}}}$. In problems of the kind contemplated this rule, the measure of the area of an isosceles triangle is given, and the value of half the base chosen at option is also given. The measures of the perpendicular and the side are then easily derived from these known quantities.