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Question Number 208327 by efronzo1 last updated on 12/Jun/24

	Answered by A5T last updated on 12/Jun/24

	$D E = 5 x, D F = 12 x$ $L e t t h e p e r p e n d i c u l a r f r o m A t o B C m e e t i t a t H;$ $A H \times B C = 3 \times 4 = 12 \Rightarrow A H = \frac{12}{5}$ $\Rightarrow B H = \sqrt{9 - \frac{144}{25}} = \frac{9}{5}$ $\frac{A D}{A B} = \frac{D E}{B C} = x \Rightarrow A D = 3 x \Rightarrow A E = 4 x$ $B F \times B C = B D \times B A \Rightarrow B F = \frac{3 \times (3 - 3 x)}{5} . . . (i)$ $\frac{B F}{B H} = \frac{D F}{A H} \Rightarrow B F = \frac{\frac{9 \times 12 x}{5}}{\frac{12}{5}} = 9 x . . . (i i)$ $(i) = (i i) \Rightarrow 3 - 3 x = 15 x \Rightarrow x = \frac{1}{6} \Rightarrow D F = 12 x = 2$
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