{"status": "success", "data": {"description_md": "Triangle $ABC$ has $AB=13,BC=14$ and $AC=15$. Let $P$ be the point on $\\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE?$\n\n$\\textbf{(A) }\\frac{42}5 \\qquad \\textbf{(B) }6\\sqrt2 \\qquad \\textbf{(C) }\\frac{84}5\\qquad \\textbf{(D) }12\\sqrt2 \\qquad \\textbf{(E) }18$\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\">AB=13,BC=14</span>  and  <span class=\"katex--inline\">AC=15</span> . Let  <span class=\"katex--inline\">P</span>  be the point on  <span class=\"katex--inline\">\\overline{AC}</span>  such that  <span class=\"katex--inline\">PC=10</span> . There are exactly two points  <span class=\"katex--inline\">D</span>  and  <span class=\"katex--inline\">E</span>  on line  <span class=\"katex--inline\">BP</span>  such that quadrilaterals  <span class=\"katex--inline\">ABCD</span>  and  <span class=\"katex--inline\">ABCE</span>  are trapezoids. What is the distance  <span class=\"katex--inline\">DE?</span> </p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }\\frac{42}5 \\qquad \\textbf{(B) }6\\sqrt2 \\qquad \\textbf{(C) }\\frac{84}5\\qquad \\textbf{(D) }12\\sqrt2 \\qquad \\textbf{(E) }18</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": "2021 AMC 12B Problem 11", "can_next": true, "can_prev": true, "nxt": "/problem/21_amc12B_p12", "prev": "/problem/21_amc12B_p10"}}