某校七年级组织学生参加数学实践活动,需将学生分成若干小组进行实地测量。已知若每组安排5人,则最后剩下3人无法编组;若每组安排7人,则最后一组只有4人。现决定重新分组,要求每组人数相同且不少于6人,不多于10人,并且所有学生恰好分完。已知学生总人数在80到120之间,求该校七年级参加活动的学生总人数,并列出所有可能的分组方案(每组人数和对应的组数)。
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题目中提到对角线互相垂直,这是菱形的重要性质之一。虽然正方形和菱形的对角线都互相垂直,但题目中给出的对角线长度分别为6和8,不相等,因此不可能是正方形(正方形对角线相等)。矩形和普通平行四边形的对角线一般不垂直,除非是特殊情况(如正方形),但此处不符合。此外,题目指出由对角线分割出的两个直角三角形斜边相等,结合对角线互相垂直,可推断四边形的四条边长度相等(因为每个边都是直角三角形的斜边,且对应直角边组合相同),进一步支持该四边形为菱形。因此,最可能的图形是菱形。
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[{"id":2416,"content":"如图,在平面直角坐标系中,点 A(1, 2)、B(4, 6)、C(7, 2) 构成三角形 ABC。若点 D 是点 A 关于直线 BC 的对称点,则点 D 的坐标最接近下列哪一项?(提示:可利用轴对称性质与一次函数求对称点)","type":"选择题","subject":"数学","grade":"八年级","stage":"初中","difficulty":"中等","answer":"C","explanation":"本题综合考查轴对称、一次函数、勾股定理与坐标几何知识。首先求直线 BC 的解析式:B(4,6)、C(7,2),斜率 k = (2−6)\/(7−4) = −4\/3,得直线 BC:y − 6 = −4\/3(x − 4),即 y = −(4\/3)x + 34\/3。点 A(1,2) 关于该直线的对称点 D 满足:AD 的中点在 BC 上,且 AD ⊥ BC。设 D(x,y),则中点 M((1+x)\/2, (2+y)\/2) 在 BC 上,代入直线方程得 (2+y)\/2 = −(4\/3)·((1+x)\/2) + 34\/3。又因 AD 斜率为 (y−2)\/(x−1),应与 BC 斜率 −4\/3 互为负倒数,即 (y−2)\/(x−1) = 3\/4。联立两个方程解得 x ≈ 11,y ≈ 4。因此点 D 坐标最接近 (11, 4)。","options":[{"id":"A","content":"(9, 6)"},{"id":"B","content":"(10, 5)"},{"id":"C","content":"(11, 4)"},{"id":"D","content":"(12, 3)"}]},{"id":2776,"content":"高中学段示例题目","type":"选择题","subject":"通用","grade":"高一","stage":"高中","difficulty":"中等","answer":"示例答案","explanation":"示例解析","options":[]},{"id":721,"content":"在一次班级图书整理活动中,某学生统计了同学们捐赠的图书数量,发现捐赠数量最多的比最少的多8本,而最多的数量是最少的3倍。如果最少捐赠了___本书,那么最多捐赠了___本书。","type":"填空题","subject":"数学","grade":"初一","stage":"初中","difficulty":"简单","answer":"4, 12","explanation":"设最少捐赠了x本书,则最多捐赠了3x本书。根据题意,最多比最少多8本,可列方程:3x - x = 8,解得2x = 8,x = 4。因此最少捐赠了4本,最多捐赠了3 × 4 = 12本。本题考查一元一次方程的实际应用,属于简单难度。","options":[]},{"id":2425,"content":"某学生测量了一个四边形的两条对角线长度分别为6 cm和8 cm,且两条对角线互相垂直。若该四边形的一组对边分别与两条对角线平行,则这个四边形的面积是( )","type":"选择题","subject":"数学","grade":"八年级","stage":"初中","difficulty":"中等","answer":"B","explanation":"根据题意,四边形的两条对角线互相垂直,长度分别为6 cm和8 cm。当四边形的对角线互相垂直时,其面积公式为:面积 = (1\/2) × 对角线₁ × 对角线₂。代入数据得:面积 = (1\/2) × 6 × 8 = 24 cm²。题目中补充条件“一组对边分别与两条对角线平行”,说明该四边形为菱形或更一般的对角线互相垂直的四边形(如筝形),但不影响面积公式的适用性,因为只要对角线互相垂直,面积公式即成立。因此正确答案为B。","options":[{"id":"A","content":"12 cm²"},{"id":"B","content":"24 cm²"},{"id":"C","content":"36 cm²"},{"id":"D","content":"48 cm²"}]},{"id":261,"content":"某学生在解方程时,将方程 3(x - 2) + 5 = 2x + 7 的括号展开后得到 3x - 6 + 5 = 2x + 7,合并同类项后得到 3x - 1 = 2x + 7。接下来,该学生将含 x 的项移到等式左边,常数项移到右边,得到 ___ = 8,解得 x = 8。","type":"填空题","subject":"数学","grade":"初一","stage":"初中","difficulty":"中等","answer":"x","explanation":"从步骤 3x - 1 = 2x + 7 开始,将 2x 移到左边变为 -2x,将 -1 移到右边变为 +1,得到 3x - 2x = 7 + 1,即 x = 8。因此,空格处应填写的是变量 x,表示移项后得到的方程是 x = 8。此题考查一元一次方程的移项与合并同类项能力,属于七年级代数基础内容。","options":[]},{"id":509,"content":"某班级组织了一次环保活动,收集废旧纸张。第一周收集了总量的40%,第二周收集了30千克,此时已收集的与未收集的质量比为3:2。问这批废旧纸张的总质量是多少千克?","type":"选择题","subject":"数学","grade":"初一","stage":"初中","difficulty":"简单","answer":"D","explanation":"设这批废旧纸张的总质量为x千克。第一周收集了40%即0.4x千克,第二周收集了30千克,因此已收集的总量为0.4x + 30千克。未收集的部分为x - (0.4x + 30) = 0.6x - 30千克。根据题意,已收集与未收集的质量比为3:2,可列方程:(0.4x + 30) \/ (0.6x - 30) = 3 \/ 2。交叉相乘得:2(0.4x + 30) = 3(0.6x - 30),即0.8x + 60 = 1.8x - 90。移项整理得:60 + 90 = 1.8x - 0.8x,即150 = x。因此总质量为150千克,正确答案为D。","options":[{"id":"A","content":"75千克"},{"id":"B","content":"100千克"},{"id":"C","content":"120千克"},{"id":"D","content":"150千克"}]},{"id":476,"content":"在一次班级环保活动中,某学生收集了可回收垃圾的重量记录如下:纸类3.5千克,塑料比纸类少1.2千克,金属是塑料重量的2倍。请问该学生收集的金属垃圾重量是多少千克?","type":"选择题","subject":"数学","grade":"初一","stage":"初中","difficulty":"简单","answer":"A","explanation":"首先根据题意,纸类重量为3.5千克。塑料比纸类少1.2千克,因此塑料重量为:3.5 - 1.2 = 2.3(千克)。金属是塑料重量的2倍,所以金属重量为:2.3 × 2 = 4.6(千克)。因此正确答案是A。本题考查有理数的加减与乘法运算在实际问题中的应用,符合七年级有理数章节的学习要求。","options":[{"id":"A","content":"4.6"},{"id":"B","content":"5.8"},{"id":"C","content":"6.4"},{"id":"D","content":"7.0"}]},{"id":204,"content":"某学生在计算一个数减去3.5时,误将减号看成了加号,结果得到8.2。正确的计算结果应该是____。","type":"填空题","subject":"数学","grade":"初一","stage":"小学","difficulty":"简单","answer":"1.2","explanation":"该学生误将减法做成了加法,即把原数加上3.5得到了8.2。因此可以先求出原数:8.2 - 3.5 = 4.7。然后用正确的运算方式计算:4.7 - 3.5 = 1.2。所以正确答案是1.2。","options":[]},{"id":1936,"content":"某学生在平面直角坐标系中绘制了一个等腰三角形ABC,顶点A的坐标为(2, 5),底边BC的两个端点B和C分别位于x轴上,且关于直线x = 2对称。若三角形ABC的面积为12,则点B的横坐标为____。","type":"填空题","subject":"数学","grade":"七年级","stage":"初中","difficulty":"困难","answer":"-1","explanation":"设B(2-a,0),C(2+a,0),则底边BC=2a,高为5。由面积公式½×2a×5=15,得a=3,故B横坐标为2-3=-1。","options":[]},{"id":238,"content":"某学生在计算一个数的相反数时,误将该数加上了3,结果得到5。那么这个数的正确相反数应该是____。","type":"填空题","subject":"数学","grade":"初一","stage":"小学","difficulty":"简单","answer":"-2","explanation":"设这个数为x。根据题意,某学生误将x加上3得到5,即x + 3 = 5,解得x = 2。这个数的相反数是-2。因此,正确答案是-2。","options":[]}]