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Question Number 182542 by cortano1 last updated on 11/Dec/22

Commented by cortano1 last updated on 11/Dec/22

$Radius semi circle PQR = 24 cm$ $radius semi circle BML = 21 cm$ $find the area ABCD .$
	Answered by mr W last updated on 11/Dec/22

	Commented by mr W last updated on 11/Dec/22

	$\frac{x - r_{1}}{r_{1}} = \frac{r_{1}}{y - r_{1}}$ $\Rightarrow x = \frac{r_{1} y}{y - r_{1}} . . . (i)$ $A C = y + \sqrt{{(x - r_{2})}^{2} - r_{2}^{2}} = \sqrt{x^{2} + y^{2}}$ $y^{2} + x (x - 2 r_{2}) + 2 \sqrt{x (x - 2 r_{2})} = x^{2} + y^{2}$ $y \sqrt{x (x - 2 r_{2})} = r_{2} x$ $y^{2} (x - 2 r_{2}) = r_{2}^{2} x$ $\Rightarrow x = \frac{2 r_{2} y^{2}}{y^{2} - r_{2}^{2}} . . . (i i)$ $\frac{2 r_{2} y^{2}}{y^{2} - r_{2}^{2}} = \frac{r_{1} y}{y - r_{1}}$ $(2 r_{2} - r_{1}) y^{2} - 2 r_{1} r_{2} y + r_{1} r_{2}^{2} = 0$ $\Rightarrow y = \frac{r_{1} r_{2} + r_{2} \sqrt{2 r_{1} (r_{1} - r_{2})}}{2 r_{2} - r_{1}}$ $= \frac{24 \times 21 + 21 \sqrt{2 \times 24 (24 - 21)}}{2 \times 21 - 24}$ $= 42 ✓$ $\Rightarrow x = \frac{24 \times 42}{42 - 24} = 56 ✓$ $a r e a o f r e c t a n g l e x y = 2352$
	Commented by cortano1 last updated on 11/Dec/22

	$nice . . .$
	Commented by EpicMaths last updated on 11/Dec/22


	Commented by Rasheed.Sindhi last updated on 12/Dec/22

	$S o m e t h i n g i s w r i t t e n i n w h i t e i n k!$
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