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triangle ABC is a right angled triangle with AB = 12cm and AC = 13cm. A circle with centre O has been inscribed inside the triangle. calculate the radius of the inscribed circle​

Answer» <html><body><p><strong>Step-by-step explanation:</strong></p><p></p><p>In △ABC,</p><p></p><p>⇒ ∠B=90 </p><p>o</p><p> [ Given ]</p><p></p><p>⇒ <a href="https://interviewquestions.tuteehub.com/tag/ab-360636" style="font-weight:bold;" target="_blank" title="Click to know more about AB">AB</a>=12cm and <a href="https://interviewquestions.tuteehub.com/tag/ac-361271" style="font-weight:bold;" target="_blank" title="Click to know more about AC">AC</a>=13cm [ Given ]</p><p></p><p>Here, O is center of a circle and x is a radius.</p><p></p><p>⇒ (AC) </p><p><a href="https://interviewquestions.tuteehub.com/tag/2-283658" style="font-weight:bold;" target="_blank" title="Click to know more about 2">2</a></p><p> =(AB) </p><p>2</p><p> +(BC) </p><p>2</p><p> [ By Pythagoras theorem ]</p><p></p><p>⇒ (13) </p><p>2</p><p> =(12) </p><p>2</p><p> +(BC) </p><p>2</p><p> </p><p></p><p>⇒ 169=144+(BC) </p><p>2</p><p> </p><p></p><p>⇒ (BC) </p><p>2</p><p> =25</p><p></p><p>∴ BC=5cm</p><p></p><p>Now, AB,BC and CA are tangents to the circle at P,N and M respectively.</p><p></p><p>∴ OP=ON=OM=x [ Radius of a circle ]</p><p></p><p>⇒ <a href="https://interviewquestions.tuteehub.com/tag/area-13372" style="font-weight:bold;" target="_blank" title="Click to know more about AREA">AREA</a> of △ABC= </p><p>2</p><p>1</p><p> </p><p> ×BC×AB</p><p> </p><p> = </p><p>2</p><p>1</p><p> </p><p> ×5×12</p><p></p><p> =30cm </p><p>2</p><p> </p><p></p><p>Area of △ABC= Area of △OAB+ Area of △OBC+ Area of △OCA</p><p></p><p>⇒ 30= </p><p>2</p><p>1</p><p> </p><p> x×AB+ </p><p>2</p><p>1</p><p> </p><p> x×BC+ </p><p>2</p><p>1</p><p> </p><p> x×CA</p><p></p><p>⇒ 30= </p><p>2</p><p>1</p><p> </p><p> x(AB+BC+CA)</p><p></p><p>⇒ x= </p><p>AB+BC+CA</p><p>2×30</p><p> </p><p> </p><p></p><p>⇒ x= </p><p>12+5+13</p><p>60</p><p> </p><p> </p><p></p><p>⇒ x= </p><p>30</p><p>60</p><p> </p><p> </p><p></p><p>∴ x=2cm </p></body></html>


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