Determine 82724

A right triangle has an area of 36 cm². A square is placed in it so that two sides of the square are parts of two sides of a triangle, and one vertex of the square is in a third of the longest side.
Determine the content of this square.

Correct answer:

S₂ =  9.6214 cm²

Step-by-step explanation:

$$\begin{gathered} S_1=36{\text{cm}}^2 \\ {S}_1=\frac{ab}{2} \\ ab=2\cdot S_1=72{\text{cm}}^2 \\ b=72/a \\ {c}^2=a^2+b^2 \\ (c/3{)}^2=(a-x{)}^2+x^2 \\ {c}^2=a^2+(72/a{)}^2 \\ (c/3{)}^2=(a-x{)}^2+x^2 \\ {c}^2=a^2+(72/a{)}^2 \\ (c/3{)}^2=(a-x{)}^2+x^2 \\ a=6.20368\doteq 6.2037\text{cm} \\ c=13.16\text{cm} \\ x=3.10184\doteq 3.1018\text{cm} \\ b=72/a=72/6.2037\doteq 11.606\text{cm} \\ {S}_2=x^2=3.1018^2\doteq 9.6214{\text{cm}}^2 \\ \text{ Verifying Solution: } \\ {c}_2=\sqrt{a^2+b^2}=\sqrt{6.2037^2+11.606^2}\doteq 13.16\text{cm} \\ {S}_3=\frac{a\cdot b}{2}=\frac{6.2037\cdot 11.606}{2}=36{\text{cm}}^2 \end{gathered}$$

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