Two arcs having their centers on a circle are cutting each other at a single point inside the circle and thus dividing the circle in four regions. If the arcs cut each other in a:b & c:d ratios what is the ratio between four regions of the circle when the circle has radius R,the arc divided in a:b has radius r_1 and the arc divided in c:d has radius r


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Question Number 68761 by Rasheed.Sindhi last updated on 15/Sep/19

$Two arcs having their centers on a$ $circle are cutting each other at a$ $single point inside the circle and thus$ $dividing the circle in four regions .$ $If the arcs cut each other in a : b & c : d$ $ratios what is the ratio between four$ $regions of the circle when the circle$ $has radius R, the arc divided in a : b$ $has radius r_{1} and the arc divided in$ $c : d has radius r_{2} .$
Commented by Rasheed.Sindhi last updated on 15/Sep/19

	Answered by mr W last updated on 18/Sep/19

	Commented by mr W last updated on 18/Sep/19

	$\cos \frac{α + β}{2} = \frac{r_{1}}{2 R}$ $\Rightarrow α + β = 2 \cos^{- 1} \frac{r_{1}}{2 R}$ $\frac{α}{β} = \frac{a}{b}$ $\Rightarrow α = \frac{2 a}{a + b} \times \cos^{- 1} \frac{r_{1}}{2 R}$ $\Rightarrow β = \frac{2 b}{a + b} \times \cos^{- 1} \frac{r_{1}}{2 R}$ $\Rightarrow γ + δ = 2 \cos^{- 1} \frac{r_{2}}{2 R}$ $\Rightarrow γ = \frac{2 c}{c + d} \times \cos^{- 1} \frac{r_{2}}{2 R}$ $\Rightarrow δ = \frac{2 d}{c + d} \times \cos^{- 1} \frac{r_{2}}{2 R}$ $l e t A_{1} = a r e a o f p a r t I E G$ $. . . . . .$
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