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Question Number 80675 by naka3546 last updated on 05/Feb/20

Commented by naka3546 last updated on 05/Feb/20

$$AreaoftriangleABC=...$$

Answered by mr W last updated on 05/Feb/20

Commented by mr W last updated on 05/Feb/20

$$\frac{x+48+z}{x}={\left(\frac{BL}{PQ}\right)}^2={\left(\frac{PQ+ML}{PQ}\right)}^2={(1+\frac{ML}{PQ})}^2$$$$1+\frac{48}{x}+\frac{z}{x}={(1+\sqrt{\frac{z}{x}})}^2=1+2\sqrt{\frac{z}{x}}+\frac{z}{x}$$$$\frac{48}{2}=\sqrt{xz}...\left(i\right)$$$$similarly$$$$\frac{32}{2}=\sqrt{zy}...\left(ii\right)$$$$\frac{96}{2}=\sqrt{yx}...\left(iii\right)$$$$\left(i\right)\times \left(ii\right)\times \left(iii\right):$$$$\frac{48}{2}\times \frac{32}{2}\times \frac{96}{2}=xyz...\left(iv\right)$$$$\left(iv\right)/{\left(i\right)}^2:$$$$y=\frac{\frac{48}{2}\times \frac{32}{2}\times \frac{96}{2}}{{\left(\frac{48}{2}\right)}^2}=\frac{\frac{32}{2}\times \frac{96}{2}}{\frac{48}{2}}=\frac{32\times 96}{2\times 48}=32$$$$similarly$$$$x=\frac{48\times 96}{2\times 32}=72$$$$z=\frac{32\times 48}{2\times 96}=8$$$$$$$${\mathrm{\Delta }}_{ABC}=48+32+96+72+32+8=288{cm}^2$$

Answered by mr W last updated on 05/Feb/20

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