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Question Number 187774 by normans last updated on 21/Feb/23

	Answered by mr W last updated on 21/Feb/23

	Commented by mr W last updated on 21/Feb/23

	$M e t h o d I$ $a = s i d e l e n g t h o f r e g u l a r h e x a g o n$ $l e t O P = d$ $\sin α = \frac{h_{1}}{d}$ $\sin β = \frac{h_{3}}{d}$ $α + β = \frac{π}{3}$ $\cos (α + β) = \cos \frac{π}{3} = \frac{1}{2}$ $\frac{\sqrt{(d^{2} - h_{1}^{2}) (d^{2} - h_{3}^{2})}}{d^{2}} - \frac{h_{1} h_{3}}{d^{2}} = \frac{1}{2}$ $(d^{2} - h_{1}^{2}) (d^{2} - h_{3}^{2}) = {(\frac{d^{2}}{2} + h_{1} h_{3})}^{2}$ $\Rightarrow 3 d^{2} = 4 (h_{1}^{2} + h_{3}^{2} + h_{1} h_{3})$ $\cos (α - \frac{π}{6}) = \frac{h_{2} + \frac{\sqrt{3} a}{2}}{d}$ $\sqrt{3} \cos α + \sin α = \frac{2 h_{2} + \sqrt{3} a}{d}$ $\frac{\sqrt{3 (d^{2} - h_{1}^{2})}}{d} + \frac{h_{1}}{d} = \frac{2 h_{2} + \sqrt{3} a}{d}$ $\sqrt{3 (d^{2} - h_{1}^{2})} + h_{1} = 2 h_{2} + \sqrt{3} a$ $\sqrt{4 (h_{1}^{2} + h_{3}^{2} + h_{1} h_{3}) - 3 h_{1}^{2}} + h_{1} = 2 h_{2} + \sqrt{3} a$ $h_{1} + 2 h_{3} + h_{1} = 2 h_{2} + \sqrt{3} a$ $\Rightarrow \frac{\sqrt{3} a}{2} = h_{1} + h_{3} - h_{2}$ $s i m i l a r l y$ $\Rightarrow \frac{\sqrt{3} a}{2} = h_{4} + h_{2} - h_{3}$ $\Rightarrow h_{4} + h_{2} - h_{3} = h_{1} + h_{3} - h_{2}$ $\Rightarrow h_{4} = h_{1} + 2 (h_{3} - h_{2})$ $\Rightarrow h = h_{1} + h_{4} = 2 (h_{1} + h_{3} - h_{2})$ $\Rightarrow X + A_{2} = 2 (A_{1} + A_{3} - A_{2})$ $\Rightarrow X + 7 = 2 (10 + 3 - 7) = 12$ $\Rightarrow X = 5$
	Answered by mr W last updated on 21/Feb/23

	Commented by mr W last updated on 21/Feb/23

	$M e t h o d I I$ $s a y P (k, h_{3})$ $e q n . l i n e L 1 :$ $y = \sqrt{3} x$ $h_{1} = \frac{\sqrt{3} k - h_{3}}{2}$ $\Rightarrow \sqrt{3} k = 2 h_{1} + h_{3}$ $e q n . l i n e L 2 :$ $y = - \sqrt{3} (x - a)$ $h_{2} = \frac{\sqrt{3} k + h_{3} - \sqrt{3} a}{2}$ $2 h_{2} = \sqrt{3} k + h_{3} - \sqrt{3} a$ $\Rightarrow \sqrt{3} a = 2 (h_{1} + h_{3} - h_{2})$ $e q n . o f l i n e L 4 :$ $y = \sqrt{3} (x - 2 a)$ $h_{4} = - \frac{\sqrt{3} k - h_{3} - 2 a \sqrt{3}}{2}$ $h_{4} = - \frac{2 h_{1} + h_{3} - h_{3} - 4 (h_{1} + h_{3} - h_{2})}{2}$ $h_{4} = h_{1} + 2 (h_{3} - h_{2})$ $h = h_{1} + h_{4} = 2 (h_{1} + h_{3} - h_{2})$ $X + A_{2} = 2 (A_{1} + A_{3} - A_{2})$ $X + 7 = 2 (10 + 3 - 7) = 12$ $X = 5$
	Commented by mr W last updated on 21/Feb/23

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