{"status": "success", "data": {"description_md": "A square of area $2$ is inscribed in a square of area $3$, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?<br><center><img class=\"problem-image\" alt=\"[asy] size(200); defaultpen(linewidth(0.6pt)+fontsize(10pt)); real y = sqrt(3); pair A,B,C,D,E,F,G,H; A = (0,0); B = (0,y); C = (y,y); D = (y,0); E = ((y + 1)/2,y); F = (y, (y - 1)/2); G = ((y - 1)/2, 0); H = (0,(y + 1)/2); fill(H--B--E--cycle, gray); draw(A--B--C--D--cycle); draw(E--F--G--H--cycle); [/asy]\" class=\"latexcenter\" height=\"335\" src=\"https://latex.artofproblemsolving.com/7/6/3/763eea423cc19d0e0a602d3e907bd5edc03cc6b7.png\" width=\"335\"/></center>\n\n$\\textbf{(A) }\\frac15\\qquad\\textbf{(B) }\\frac14\\qquad\\textbf{(C) }2-\\sqrt3\\qquad\\textbf{(D) }\\sqrt3-\\sqrt2\\qquad\\textbf{(E) }\\sqrt2-1$\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>A square of area  <span class=\"katex--inline\">2</span>  is inscribed in a square of area  <span class=\"katex--inline\">3</span> , creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?<br/><center><img class=\"latexcenter\" alt=\"[asy] size(200); defaultpen(linewidth(0.6pt)+fontsize(10pt)); real y = sqrt(3); pair A,B,C,D,E,F,G,H; A = (0,0); B = (0,y); C = (y,y); D = (y,0); E = ((y + 1)/2,y); F = (y, (y - 1)/2); G = ((y - 1)/2, 0); H = (0,(y + 1)/2); fill(H--B--E--cycle, gray); draw(A--B--C--D--cycle); draw(E--F--G--H--cycle); [/asy]\" height=\"335\" src=\"https://latex.artofproblemsolving.com/7/6/3/763eea423cc19d0e0a602d3e907bd5edc03cc6b7.png\" width=\"335\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }\\frac15\\qquad\\textbf{(B) }\\frac14\\qquad\\textbf{(C) }2-\\sqrt3\\qquad\\textbf{(D) }\\sqrt3-\\sqrt2\\qquad\\textbf{(E) }\\sqrt2-1</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": 2, "problem_name": "2023 AMC 12A Problem 9", "can_next": true, "can_prev": true, "nxt": "/problem/23_amc12A_p10", "prev": "/problem/23_amc12A_p08"}}