in the triangle ABC, angle ABC=90degree,AB=2a+1 and BC=2a*a+2a.Find AC in terms of a.If a=8 find the lengths of the sides of the triangle ABC.

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$$\begin{gathered} in a right angled triangle \\ Given \\ Side AB=2a+1 \\ BC=2a^2+2a=2a\left(a+1\right) \\ find AC \\ U\sin g Pythagoras theorem \\ A{C}^2=A{B}^2+B{C}^2 \\ AC=\sqrt{A{B}^2+B{C}^2} \\ AC=\sqrt{{\left(2a+1\right)}^2+{\left(2a\left(a+1\right)\right)}^2} \\ AC=\sqrt{4a^2+1+4a+\left[4a^2\left(a^2+1+2a\right)\right]} \\ AC=\sqrt{4a^2+1+4a+4a^4+4a^2+8a^3} \\ \mathit{A}\mathit{C}\mathbf{=}\sqrt{\mathbf{4}\mathbf{a}\mathbf{+}\mathbf{1}\mathbf{+}\mathbf{4}{\mathbf{a}}^{\mathbf{4}}\mathbf{+}\mathbf{8}{\mathbf{a}}^{\mathbf{2}}\mathbf{+}\mathbf{8}{\mathbf{a}}^{\mathbf{3}}} \\ Given If a=8 \\ then \\ AB=2a+1 \\ =2\times 8+1 \\ \mathit{A}\mathit{B}\mathbf{ }\mathbf{=}\mathbf{17}\mathbf{ }\mathit{u}\mathit{n}\mathit{i}\mathit{t} \\ BC=2a^2+2a \\ =2\times 8^2+2\times 8 \\ =2\times 64+16 \\ =128+16 \\ \mathit{B}\mathit{C}\mathbf{=}\mathbf{144}\mathbf{ }\mathit{u}\mathit{n}\mathit{i}\mathit{t} \\ \mathbf{ }AC=\sqrt{4a+1+4a^4+8a^2+8a^3} \\ AC=\sqrt{4\times 8+1+4\times 8^4+8\times 8^2+8\times 8^3} \\ AC=\sqrt{32+1+16384+512+4096} \\ AC=\sqrt{21025} \\ \mathit{A}\mathit{C}\mathbf{=}\mathbf{145}\mathit{u}\mathit{n}\mathit{i}\mathit{t} \\ \mathit{A}\mathit{n}\mathit{s}\mathit{w}\mathit{e}\mathbf{ }\mathbf{ }\mathbf{ }\mathit{A}\mathit{C}\mathbf{=}\sqrt{\mathbf{4}\mathbf{a}\mathbf{+}\mathbf{1}\mathbf{+}\mathbf{4}{\mathbf{a}}^{\mathbf{4}}\mathbf{+}\mathbf{8}{\mathbf{a}}^{\mathbf{2}}\mathbf{+}\mathbf{8}{\mathbf{a}}^{\mathbf{3}}}\mathbf{ }\mathbf{ } \\ \mathit{w}\mathit{h}\mathit{e}\mathit{n}\mathbf{ }\mathit{a}\mathbf{=}\mathbf{8}\mathbf{ }\mathbf{ }\mathbf{;}\mathbf{ }\mathit{A}\mathit{B}\mathbf{=}\mathbf{17}\mathit{u}\mathit{n}\mathit{i}\mathit{t}\mathbf{ }\mathbf{,}\mathbf{ }\mathit{B}\mathit{C}\mathbf{=}\mathbf{144}\mathbf{ }\mathit{u}\mathit{n}\mathit{i}\mathit{t}\mathbf{,}\mathbf{ }\mathit{A}\mathit{C}\mathbf{=}\mathbf{145}\mathbf{ }\mathit{u}\mathit{n}\mathit{i}\mathit{t} \end{gathered}$$


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