Given a triangle with a 112^circ angle. The adjusted sides to this angle are equal. Find their length, if A=80cm^2.

Question
Given a triangle with a \(\displaystyle{112}^{\circ}\) angle. The adjusted sides to this angle are equal. Find their length, if \(\displaystyle{A}={80}{c}{m}^{{2}}\).

Answers (1)

2021-02-04
Let each of the sides be S.
The formula for the triangle with equal sides:
\(\displaystyle{A}=\frac{{1}}{{2}}{S}^{{2}}{{\sin}^{{2}}\theta}\)
Isolate S:
\(\displaystyle{S}=\sqrt{{\frac{{{2}{A}}}{{{{\sin}^{{2}}\theta}}}}}\)
Substitute \(\displaystyle\theta={112}^{\circ}{\quad\text{and}\quad}{A}={80}{c}{m}^{{2}}\) from the given:
\(\displaystyle{S}=\sqrt{{\frac{{{160}}}{{{{\sin}^{{2}}{\left({112}^{\circ}\right)}}}}}}\)
\(\displaystyle=\frac{{{4}\sqrt{{10}}}}{{\sqrt{{{{\sin}^{{2}}{\left({112}^{\circ}\right)}}}}}}\)
\(\displaystyle=\frac{{{4}\sqrt{{10}}\sqrt{{{{\sin}^{{2}}{\left({112}^{\circ}\right)}}}}}}{{\sqrt{{{{\sin}^{{2}}{\left({112}^{\circ}\right)}}}}\sqrt{{{{\sin}^{{2}}{\left({112}^{\circ}\right)}}}}}}\)
\(\displaystyle=\frac{{{4}\sqrt{{10}}\sqrt{{{{\sin}^{{2}}{\left({112}^{\circ}\right)}}}}}}{{{{\sin}^{{2}}{\left({112}^{\circ}\right)}}}}\)
=13.64250...
0

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