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Question Number 72734 by Mr Jor last updated on 01/Nov/19

$$AtriangleABCisinscribedina$$$$circle.AC=10cm,BC=7cmand$$$$AB=10cm.Findtheradiusofthe$$$$circle.$$

Answered by $@ty@m123 last updated on 01/Nov/19

$$a=7,b=10,c=10$$$$s=\frac{a+b+c}{2}=\frac{27}{2}$$$$△=\sqrt{s(s-a)(s-b)(s-c)}$$$$=\sqrt{\frac{27}{2}(\frac{27}{2}-7)(\frac{27}{2}-10)(\frac{27}{2}-10)}$$$$=\sqrt{\frac{27}{2}\times \frac{13}{2}\times \frac{7}{2}\times \frac{7}{2}}$$$$=\frac{21}{4}\sqrt{39}......\left(1\right)$$$$R=\frac{abc}{4△}$$$$=\frac{7\times 10\times 10}{4\times \frac{21\sqrt{39}}{4}}$$$$=\frac{100}{3\sqrt{39}}Ans.$$

Answered by MJS last updated on 01/Nov/19

$$\mathrm{triangle}\mathrm{with}\mathrm{side}\mathrm{lengths}a,b,c$$$$\mathrm{circumcircle}R=\frac{abc}{\delta }$$$$\delta =\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}$$

Commented by $@ty@m123 last updated on 01/Nov/19

$$a+b+c=2s$$$$\therefore \delta =\sqrt{2s.(2s-2a)(2s-2b)(2s-2c)}$$$$=4\sqrt{s(s-a)(s-b)(s-c)}$$$$=4△$$

Answered by mr W last updated on 01/Nov/19

$$\sqrt{{10}^2-{\left(\frac{7}{2}\right)}^2}-R=\sqrt{R^2-{\left(\frac{7}{2}\right)}^2}$$$${10}^2-2R\sqrt{{10}^2-{\left(\frac{7}{2}\right)}^2}=0$$$$\Rightarrow R=\frac{{10}^2}{\sqrt{{\left(2\times 10\right)}^2-7^2}}=\frac{100}{\sqrt{27\times 13}}=\frac{100\sqrt{39}}{117}\approx 5.338$$

Commented by mr W last updated on 01/Nov/19

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