{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"math","formal_name":"MATH","introduction":"MATH organises 12,500 high-school competition problems into seven subjects and five difficulty levels, each with a full worked solution. It underlies much of the later mathematics evaluation work; this catalogue imports the full test split: 5,000 problems across all seven subjects.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://github.com/hendrycks/math","indexing_mode":"index","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[],"history":"","construction":"","limitations":"","saturation":"","scores":[]}},"task_id":"ebc8c9ed-0149-572a-999e-e47d9a1e5cd5","task_key":"geometry--test--ebc8c9ed-0149-572a-999e-e47d9a1e5cd5","task_revision_id":"1","upstream_id":"","short_description":"In rectangle $ABCD$ with $AB = 16,$ $P$ is a point on $BC$ so that $\\angle…","config":"geometry","split":"test","body":"{\"level\":\"Level 4\",\"problem\":\"In rectangle $ABCD$ with $AB = 16,$ $P$ is a point on $BC$ so that $\\\\angle APD=90^{\\\\circ}$. $TS$ is perpendicular to $BC$ with $BP=PT$, as shown.  $PD$ intersects $TS$ at $Q$.  Point $R$ is on $CD$ such that $RA$ passes through $Q$.  In $\\\\triangle PQA$, $PA=20$, $AQ=25$ and $QP=15$. [asy]\\nsize(7cm);defaultpen(fontsize(9));\\nreal sd = 7/9 * 12;\\npath extend(pair a, pair b) {return a--(10 * (b - a));}\\n\\n// Rectangle\\npair a = (0, 0); pair b = (0, 16); pair d = (24 + sd, 0); pair c = (d.x, b.y);\\ndraw(a--b--c--d--cycle);\\nlabel(\\\"$A$\\\", a, SW);label(\\\"$B$\\\", b, NW);label(\\\"$C$\\\", c, NE);label(\\\"$D$\\\", d, SE);\\n\\n// Extra points and lines\\npair q = (24, 7); pair s = (q.x, 0); pair t = (q.x, b.y);\\npair r = IP(c--d, extend(a, q));\\npair p = (12, b.y);\\ndraw(q--a--p--d--r--cycle);draw(t--s);\\nlabel(\\\"$R$\\\", r, E); label(\\\"$P$\\\", p, N);label(\\\"$Q$\\\", q, 1.2 * NE + 0.2 * N);label(\\\"$S$\\\", s, S); label(\\\"$T$\\\", t, N);\\n\\n// Right angles and tick marks\\nmarkscalefactor = 0.1;\\ndraw(rightanglemark(a, b, p)); draw(rightanglemark(p, t, s)); draw(rightanglemark(q, s, d));draw(rightanglemark(a, p, q));\\nadd(pathticks(b--p, 2, spacing=3.4, s=10));add(pathticks(p--t, 2, spacing=3.5, s=10));\\n\\n// Number labels\\nlabel(\\\"$16$\\\", midpoint(a--b), W); label(\\\"$20$\\\", midpoint(a--p), NW); label(\\\"$15$\\\", midpoint(p--q), NE);\\nlabel(\\\"$25$\\\", midpoint(a--q), 0.8 * S + E);\\n[/asy] Find $QR - RD$.\",\"type\":\"Geometry\"}","display_format":"math","language":"","answer_status":"published","assets":[],"source_url":"https://github.com/hendrycks/math","history":"initial import","indexing_mode":"index","subproblems":[],"grids":[]}