Legs (sides other than the hypotenuse) of a r | vedclass

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(B) Let $ABC$ be a right triangle right-angled at $B$ with $AB = 16 \,cm$ and $BC = 8 \,cm$. The largest square $BRSP$ that can be inscribed in this t

Vedclass · Accepted Answer

$\frac{16}{3}$

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(B) Let $ABC$ be a right triangle right-angled at $B$ with $AB = 16 \,cm$ and $BC = 8 \,cm$. The largest square $BRSP$ that can be inscribed in this triangle is shown in the figure.Let the side of the square be $x \,cm$. So,$PB = x \,cm$ and $BR = x \,cm$.Then $AP = AB - PB = (16 - x) \,cm$.In $	riangle APS$ and $	riangle ABC$,$\angle A = \angle A$ (common) and $\angle APS = \angle ABC = 90^{\circ}$.Therefore,$	riangle APS \sim 	riangle ABC$ by $AA$ similarity.This implies $\frac{AP}{AB} = \frac{PS}{BC}$.Substituting the values,$\frac{16 - x}{16} = \frac{x}{8}$.Cross-multiplying,$8(16 - x) = 16x$.$128 - 8x = 16x$.$128 = 24x$.$x = \frac{128}{24} = \frac{16}{3} \,cm$.

Legs (sides other than the hypotenuse) of a right triangle are of lengths $16 \,cm$ and $8 \,cm$. Find the length of the side of the largest square that can be inscribed in the triangle. (in $cm$)

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