如图①,在
中,
,
,
为斜边
的中点,可以发现
和
的数量关系为________.
由此,小亮猜想“直角三角形斜边上的中线等于斜边的一半.”
小莹采用了下面的证明过程,请你补充完整.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/25/5da1d50e-f6dd-46cd-ad96-df100f4149ea.png?resizew=328)
已知:如图②,在
中,
为斜边
的中点.
求证:________________.
证明:延长
至点
,使得
,连接
.
∵
为斜边
的中点(已知),
∴
(线段中点的定义).
∵
(________________),
且
(已作),
∴________________(
).
∴
且
(________________).
∴
(________________).
∴
(两直线平行,同旁内角互补)
∴
(等式的性质).
在
和
中.
∵________________,
∴
(________________).
∴
(全等三角形对应边相等).
∵
(已作),
∴
(线段中点的定义).
∴
________________(等量代换).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eddf629bfb2d7b9daea6f2c1553360af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed10df4140819d5451773a45de66201b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b10134e7a46e6f6f7cb9d5e2371727d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
由此,小亮猜想“直角三角形斜边上的中线等于斜边的一半.”
小莹采用了下面的证明过程,请你补充完整.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/25/5da1d50e-f6dd-46cd-ad96-df100f4149ea.png?resizew=328)
已知:如图②,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eddf629bfb2d7b9daea6f2c1553360af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
求证:________________.
证明:延长
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df34369163e511f14028168cb0b21186.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccaee8f228ff24e7c89879bb5b999cf2.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ff29971ccc633d89832ffa9bd54afa3.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c55e91b1c11c0b9715e758c1d30a5e2a.png)
且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df34369163e511f14028168cb0b21186.png)
∴________________(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9730d8c39620a403e38171c723d3dfea.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5348ba6e3f7711299ce35d421b3bac5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1120e471627069e78a6733af07687684.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b43be568ae5186900cfb39484a3c9301.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0090baa658a01802362b003aa28ed19.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c922224d3dc884fc798f5bcc61b0b38.png)
在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eddf629bfb2d7b9daea6f2c1553360af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d19e6695f0f24d0254d175547237200c.png)
∵________________,
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/571a4ab985b0c7643c4c4cb3b2becfe6.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17101383eb0787edaaa35adfcd20d5c9.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df34369163e511f14028168cb0b21186.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7af9343e86e503de88b3f4682647a994.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c26a46e7879436d532af3f4b6e258a81.png)
更新时间:2023-03-08 20:03:31
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