Math, asked by NotUmair, 24 days ago

Two equal sides of an isosceles triangle are 10 cm and the perimeter is 28 cm, then the area of the triangle is

Answers

Answered by Anonymous
28

Given :

  • Equal sides of triangle = 10 cm
  • Type of Triangle = Isosceles
  • Perimeter of triangle = 28 cm

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To Find :

  • Area of the Triangle = ?

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Solution :

~ Formula Used :

  • Perimeter of Triangle :

 {\color{cyan}{\bigstar}} \; {\underline{\boxed{\red{\sf{ Perimeter{\small_{(Triangle)}} = a + b + c }}}}}

  • Area of Triangle :

 {\color{cyan}{\bigstar}} \; {\underline{\boxed{\red{\sf{ Area{\small_{(Triangle)}} = \sqrt{s (s - a)(s - b)(s - c) } }}}}}

Where :

  • ➢ s = Semi - Perimeter
  • ➢ a = Side 1
  • ➢ b = Side 2
  • ➢ c = Side 3

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~ Calculating the 3rd Side :

 \begin{gathered} \dashrightarrow \sf {Perimeter = a + b + c } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \sf {28 = 10 + 10 + c } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \sf {28 = 20 + c } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \sf {28 - 20 = c } \\ \end{gathered}

 \begin{gathered} \dashrightarrow {\qquad{\pink{\sf {3rd \; Side = \; 8 \; cm }}}} \\ \end{gathered}

 \\ \qquad{\rule{150pt}{1pt}}

~ Calculating the Area :

  • Semi - Perimeter :

 \begin{gathered} \implies \sf {s = \dfrac{a + b + c}{2} } \\ \end{gathered}

 \begin{gathered} \implies \sf {s = \dfrac{10 + 10 + 8}{2} } \\ \end{gathered}

 \begin{gathered} \implies \sf {s = \dfrac{28}{2} } \\ \end{gathered}

 \begin{gathered} \implies \sf {s = \cancel\dfrac{28}{2} } \\ \end{gathered}

 \begin{gathered} \implies {\qquad{\purple{\sf {Semi - Perimeter = \; 14 \; cm  }}}} \\ \end{gathered}

  • Area :

 \begin{gathered} \dashrightarrow \sf { Area = \sqrt{s (s - a)(s - b)(s - c) } } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \sf { Area = \sqrt{14 (14 - 10)(14 - 10)(14 - 8) } } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \sf { Area = \sqrt{14 \times 4 \times 4 \times 6 } } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \sf { Area = \sqrt{2 \times 7 \times 4 \times 4 \times 3 \times 2 } } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \sf { Area = \sqrt{\underline{2} \times 7 \times \underline{4 \times 4} \times 3 \times \underline{2} } } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \sf { Area = 2 \times 4 \times \sqrt{7} \times \sqrt{3} } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \sf { Area = 8 \times \sqrt{7} \times \sqrt{3} } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \sf { Area = 8 \times \sqrt{21} } \; \; \; \; \bigg\lgroup{\color{maroon}{\sf{ Taking \; \sqrt{21} = 4.583 }}} \bigg\rgroup \\ \end{gathered}

 \begin{gathered} \dashrightarrow \sf { Area = 8 \times 4.583 } \\ \end{gathered}

 \begin{gathered} \dashrightarrow {\qquad{\orange{\sf {Area = \; 36.664 \; cm²  }}}} \\ \end{gathered}

 \\ \qquad{\rule{150pt}{1pt}}

~ Therefore :

❝ Area of the given Triangle is 36.664 cm² . ❞

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Answered by divyapakhare468
6

To find : area of isosceles triangle .

Given : two equal sides = 10\ cm

             Perimeter = 28\ cm

Solution :

  • Let , a, b , c be the sides of isosceles triangle .
  • We know that , formula for perimeter of triangle is -

        Perimeter \ of \ triangle = a + b + c\\

  • Then,

             28 = 10 + 10 + c\\c = 20 -10\\c = 8

  • We find area of triangle by Heron's Formula :

       Area \ of \ triangle =\sqrt{s ( s - a)(s - b) (s - c )}

  • Where, s = semi perimeter and ,a, b, c are sides of triangle
  • semi \ perimeter , s = \frac{28}{2}  \\

                                    = 14\ cm

  • Putting values , we get

       \text { Area }=\sqrt{\mathrm{s}(\mathrm{s}-\mathrm{a})(\mathrm{s}-\mathrm{b})(\mathrm{s}-\mathrm{c})}

  •         \begin{array}{l}=\sqrt{14(14-10)(14-10)(14-8)} \\=\sqrt{14 \times 4 \times 4 \times 6} \\=\sqrt{2 \times 7 \times 4 \times 4 \times 3 \times 2} \\=\sqrt{2 \times 7 \times 4 \times 4 \times 3 \times 2} \\=2 \times 4 \times \sqrt{7} \times \sqrt{3} \\=8 \times \sqrt{7} \times \sqrt{3}\end{array}

                 \begin{array}{l}=8 \times \sqrt{21} \\=8 \times 4.583\end{array}  

        Area = 36.664 \ cm^{2}  

Hence , area of given isosceles triangle is 36.664 cm^{2}

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