{"id":388,"date":"2019-12-05T06:27:57","date_gmt":"2019-12-05T06:27:57","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=388"},"modified":"2020-10-15T06:33:02","modified_gmt":"2020-10-15T06:33:02","slug":"circles-in-a-square-part-8","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=388","title":{"rendered":"Circles in a Square – Part 8"},"content":{"rendered":"\n
Continue the same topic of the previous post in this series, a line is drawn between point\u00a0L\u00a0and the tangent point\u00a0M\u00a0between the full circle and the quarter circle, as shown in the following figure. Prove\u00a0$\\triangle{HLM}$ is a right triangle.<\/p>\n\n\n\n
<\/figure><\/div>\n\n\n\n
Based on the result of the previous post, the radius of the full circle is $\\dfrac{1}{9}$, the radius of the semi-circle is $\\dfrac{1}{4}$, we have:\n$$\\dfrac{\\overline{HM}}{\\overline{HL}}=\\dfrac{\\overline{HM}}{\\overline{KL}-\\overline{HK}}=\\dfrac{\\dfrac{1}{9}}{\\dfrac{1}{4}-\\dfrac{1}{9}}=\\dfrac{4}{5}$$\n$$\\dfrac{\\overline{HJ}}{\\overline{HC}}=\\dfrac{\\overline{KJ}-\\overline{HK}}{\\overline{HM}+\\overline{MC}}=\\dfrac{1-\\dfrac{1}{9}}{1+\\dfrac{1}{9}}=\\dfrac{4}{5}$$\nTherefore\n$$\\dfrac{\\overline{HM}}{\\overline{HL}}=\\dfrac{\\overline{HJ}}{\\overline{HC}}$$\nSince $\\triangle{HLM}$ and $\\triangle{HCJ}$ share the same angle at point $H$, $\\triangle{HLM}$ and $\\triangle{HCJ}$ are similar. Because $\\triangle{HCJ}$ is a right triangle, $\\triangle{HLM}$ is also a right triangle. \n<\/p>\n\n\n\n\n
Can you prove it without calculating the radius of the full circle? To be continued…<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"
![\"\"](\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2019-12-04-at-7.36.21-PM.png\")
<\/figure><\/div>\n\n\n\n