Question

已知：矩形紙片ABCD中，AB=26厘米，BC=18.5厘米，點E在AD上，且AE=6厘米，點P是AB邊上一動點．按如下操作：

步驟一，折疊紙片，使點P與點E重合，展開紙片得折痕MN（如圖1所示）；

步驟二，過點P作PT⊥AB，交MN所在的直線于點Q，連接QE（如圖2所示）

1.無論點P在AB邊上任何位置，都有PQ_________QE（填“”、“”、“”號）；

2.如圖3所示，將紙片ABCD放在直角坐標系中，按上述步驟一、二進行操作：

①當點P在A點時，PT與MN交于點Q₁，Q₁點的坐標是（_______，_________）；

②當PA=6厘米時，PT與MN交于點Q₂. Q₂點的坐標是（_______，_________）；

③當PA=12厘米時，在圖3中畫出MN,PT（不要求寫畫法），并求出MN與PT的交點Q₃的坐標；

3.點P在運動過程，PT與MN形成一系列的交點Q₁，Q₂，Q₃……觀察、猜想：眾多的交點形成的圖象是什么？并直接寫出該圖象的函數(shù)表達式．

 

Answer 1

【答案】

 

1.

2.；②．③畫圖見解析。

3.拋物線 、函數(shù)關(guān)系式：

【解析】（1）．······························································································· 1分

（2）①；②．······························································································· 3分

③畫圖，如圖所示．······································································································ 5分

解：方法一：設(shè)與交于點．

在中，，

．

，，

 

．

又，

．

．

．

．················································································································ 7分

方法二：過點作，垂足為，則四邊形是矩形．

，．

設(shè)，則．

在中，．

．

．

．

．

（3）這些點形成的圖象是一段拋物線．········································································ 8分

函數(shù)關(guān)系式：．····································································· 10分

說明：若考生的解答：圖象是拋物線，函數(shù)關(guān)系式：均不扣分．