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Question Number 209932 by cherokeesay last updated on 26/Jul/24

	Answered by mahdipoor last updated on 26/Jul/24

	${B D}^{2} = 100 + 49 - 140 c o s C = 121 + 64 - 176 c o s A$ $\frac{B D}{s i n C} = \frac{B D}{s i n A} = 2 R \Rightarrow C + A = 180$ $\Rightarrow 149 - 140 c o s C = 185 - 176 c o s A$ $\Rightarrow c o s C = \frac{185 - 149}{- 176 - 140} = - \frac{9}{79}$ $\Rightarrow 2 R = \frac{\sqrt{149 + 140 \times \frac{9}{79}}}{\sqrt{1 - {(\frac{9}{79})}^{2}}} = 12.93$
	Answered by a.lgnaoui last updated on 26/Jul/24
	$AB.AD=2R×h h=ABsin A1 R=((AD)/(2sin A1))=((8R)/( (√(4R^2 −121)))) 1=(8/( (√(4R^2 −121)))) 4R^2 =185 R =((√(185))/2) determinant (((AB.AD=2R.h h=ABsin A1)),( determinant ((( A1=∡ABD] )),(( R=6,8))))) Developement du resultat: BD^2 =185−176cos A A=(A1+A2)O { ((cos A1=((11)/(2R)) sin A1=(1/(2R))(√(4R^2 −121)))),((cos A2=(5/R) sin A2=(1/R)(√(R^2 −25)))) :} =149−140cos C ⇒ 36=176cos A+140cos A ⇒ 36=316cos A determinant ((),( determinant (((cos A )),(( (9/(79)) ))))) [1] sin A=sinA1cos A2+sin A2cos A1 cos A=cosA1 .cosA2 −sinA1 .sinA2 { () :}$
	$AB . AD = 2 R \times h$ $h = AB \sin A 1$ $R = \frac{AD}{2 \sin A 1} = \frac{8 R}{\sqrt{4 R^{2} - 121}}$ $1 = \frac{8}{\sqrt{4 R^{2} - 121}}$ $4 R^{2} = 185 R = \frac{\sqrt{185}}{2}$ $\begin{array}{cc} AB . AD = 2 R . h h = AB \sin A 1 \\ \begin{array}{cc} A 1 = ∡ ABD] \\ R = 6, 8 \end{array} \end{array}$ $D e v e l o p e m e n t d u r e s u l t a t :$ ${BD}^{2} = 185 - 176 \cos A A = (A 1 + A 2) O$ ${\begin{cases} \cos A 1 = \frac{11}{2 R} \sin A 1 = \frac{1}{2 R} \sqrt{{4 R}^{2} - 121} \\ \cos A 2 = \frac{5}{R} \sin A 2 = \frac{1}{R} \sqrt{R^{2} - 25} \end{cases}$ $= 149 - 140 \cos C$ $\Rightarrow 36 = 176 \cos A + 140 \cos A$ $\Rightarrow 36 = 316 \cos A$ $\begin{matrix} \begin{array}{cc} \cos A \\ \frac{9}{79} \end{array} \end{matrix} [1]$ $\sin A = sinA 1 \cos A 2 + \sin A 2 \cos A 1$ $\cos A = cosA 1 . cosA 2 - sinA 1 . sinA 2$ ${\begin{cases} \end{cases}$
	Commented by a.lgnaoui last updated on 26/Jul/24

	Answered by A5T last updated on 26/Jul/24

	$10^{2} + 11^{2} - 2 (10) (11) c o s B = 7^{2} + 8^{2} + 2 (7) (8) c o s (B)$ $\Rightarrow c o s B = \frac{27}{83} \Rightarrow s i n B = \frac{4 \sqrt{385}}{83} \Rightarrow A C = \frac{\sqrt{1029449}}{83}$ $\Rightarrow \frac{10 \times 11 \times \frac{\sqrt{1029449}}{83}}{4 R} = \frac{1}{2} \times 10 \times 11 \times \frac{4 \sqrt{385}}{83}$ $\Rightarrow R = \frac{\sqrt{396337865}}{3089} \approx 6.4637$
	Answered by mr W last updated on 26/Jul/24

	$R = \sqrt{\frac{(10 \times 11 + 7 \times 8) (10 \times 8 + 7 \times 11) (10 \times 7 + 8 \times 11)}{(- 10 + 11 + 8 + 7) (10 - 11 + 8 + 7) (10 + 11 - 8 + 7) (10 + 11 + 8 - 7)}}$ $= \frac{\sqrt{396337865}}{3080} \approx 6.4637$
	Commented by mr W last updated on 26/Jul/24

	$R = \sqrt{\frac{(a b + c d) (a c + b d) (a d + b c)}{(- a + b + c + d) (a - b + c + d) (a + b - c + d) (a + b + c - d)}}$
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