{"status": "success", "data": {"description_md": "Triangle $ABC$ has $AC=3$, $BC=4$, and $AB=5$. Point $D$ is on $\\overline{AB}$, and $\\overline{CD}$ bisects the right angle. The inscribed circles of $\\triangle ADC$ and $\\triangle BCD$ have radii $r_a$ and $r_b$, respectively. What is $r_a/r_b$?\n\n$\\mathrm{(A)}\\ \\frac{1}{28}\\left(10-\\sqrt{2}\\right)\\qquad\\mathrm{(B)}\\ \\frac{3}{56}\\left(10-\\sqrt{2}\\right)\\qquad\\mathrm{(C)}\\ \\frac{1}{14}\\left(10-\\sqrt{2}\\right)\\qquad\\mathrm{(D)}\\ \\frac{5}{56}\\left(10-\\sqrt{2}\\right)\\\\\\mathrm{(E)}\\ \\frac{3}{28}\\left(10-\\sqrt{2}\\right)$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>Triangle  <span class=\"katex--inline\">ABC</span>  has  <span class=\"katex--inline\">AC=3</span> ,  <span class=\"katex--inline\">BC=4</span> , and  <span class=\"katex--inline\">AB=5</span> . Point  <span class=\"katex--inline\">D</span>  is on  <span class=\"katex--inline\">\\overline{AB}</span> , and  <span class=\"katex--inline\">\\overline{CD}</span>  bisects the right angle. The inscribed circles of  <span class=\"katex--inline\">\\triangle ADC</span>  and  <span class=\"katex--inline\">\\triangle BCD</span>  have radii  <span class=\"katex--inline\">r_a</span>  and  <span class=\"katex--inline\">r_b</span> , respectively. What is  <span class=\"katex--inline\">r_a/r_b</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(A)}\\ \\frac{1}{28}\\left(10-\\sqrt{2}\\right)\\qquad\\mathrm{(B)}\\ \\frac{3}{56}\\left(10-\\sqrt{2}\\right)\\qquad\\mathrm{(C)}\\ \\frac{1}{14}\\left(10-\\sqrt{2}\\right)\\qquad\\mathrm{(D)}\\ \\frac{5}{56}\\left(10-\\sqrt{2}\\right)\\\\\\mathrm{(E)}\\ \\frac{3}{28}\\left(10-\\sqrt{2}\\right)</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2008 AMC 12A Problem 20", "can_next": true, "can_prev": true, "nxt": "/problem/08_amc12A_p21", "prev": "/problem/08_amc12A_p19"}}