若一个直角三角形两直角边的边长为a,b,且均为正整数,且满足a+b=m+2和ab=4m,试求直角三角形三边长 设a,b是一直角三角形的两条直角边，且a,b是方程x^2-2...
\u8fb9\u957f\u4e3a\u6574\u6570\u7684\u76f4\u89d2\u4e09\u89d2\u5f62,\u82e5\u5176\u4e24\u76f4\u89d2\u8fb9a\u3001b\u6ee1\u8db3a+b=k+2\u4e14ab=4k,\u6c42k\u3001a\u3001b\u7684\u503c\u3002\u8fd9\u9053\u9898\u76ee\u662f\u4e00\u9053\u4e0d\u9519\u7684\u7efc\u5408\u9898\u76ee\u3002\u77e5\u8bc6\u70b9\u7ed3\u5408\u7684\u6bd4\u8f83\u597d\uff0c\u800c\u4e14\u6709\u4e9b\u5185\u5bb9\u8003\u5bdf\u7684\u6bd4\u8f83\u6df1\u3002\u591a\u5e94\u4e3a\u7ade\u8d5b\u9898\u76ee\u5427\u3002
\u5df2\u77e5a+b\u548cab\uff0c\u7531\u97e6\u8fbe\u5b9a\u7406\uff0c\u53ef\u4ee5\u6784\u9020\u4e8c\u5143\u4e00\u6b21\u65b9\u7a0b\uff0cx^2-(k+4)x+4k=0,\u5176\u4e2da\u3001b\u4e3a\u65b9\u7a0b\u7684\u4e24\u4e2a\u6574\u6570\u89e3\u3002\u95ee\u9898\u7684\u5173\u952e\u5c31\u662f\u5728\u4e8e\u4e24\u4e2a\u6574\u6570\u89e3\u3002\u03b4=\uff08k+2)^2-16k=k^2-12k+4,\u7531\u4e8e\u5b58\u5728\u4e24\u4e2a\u6574\u6570\u89e3\uff0c\u6240\u4ee5\u03b4\u5fc5\u4e3a\u4e00\u4e2a\u6570\u7684\u5e73\u65b9\uff0c\u4e0d\u59a8\u8bbe\u03b4=m^2,\uff08m>0\uff09\u53ef\u4ee5\u5316\u7b80\u6b64\u7b49\u5f0f\u3002\u5c06k\u914d\u65b9,\u5229\u7528\u5e73\u65b9\u5dee\u516c\u5f0f\u53ef\u5f97(k-6-m)(k-6+m)=32\uff08\u6b64\u4e3a\u4e0d\u5b9a\u65b9\u7a0b\uff09\uff0c\u6613\u77e5\u5f0f\u4e2d\u5de6\u8fb9\u7b2c\u4e00\u9879\u5c0f\u4e8e\u7b2c\u4e8c\u9879\uff0c\u4e14\u4e24\u9879\u5747\u4e3a\u6574\u6570\uff0c\u5e76\u4e14\u540c\u5947\u5076\u53ef\u77e5k-6-m=2,k-6+m=16\u6216\u8005k-6-m=4,k-6+m=8
\u89e3\u8fd9\u4e24\u4e2a\u65b9\u7a0b\u53ef\u77e5\uff0ck=12\u621615\uff0c\u4ece\u800c\u8fdb\u4e00\u6b65\u5f97\u5230 \u5f53k=12\u65f6a=6\uff0cb=8\uff08\u6216\u8005a\u3001b\u4ea4\u6362\uff0c\u4ee5\u4e3a\u6ca1\u6709\u544a\u8bc9\u5927\u5c0f\uff09\uff0c\u5f53k=15\u65f6\uff0ca=5\uff0cb=12\uff08\u6216\u8005\u4ea4\u6362\uff09
\u8bbe\u662fa\uff0cb\uff0cc\uff0cd\u6b63\u6574\u6570\uff0ca\uff0cb\u662f\u65b9\u7a0bx2-\uff08d-c\uff09x+cd=0\u7684\u4e24\u4e2a\u6839\uff0e\u8bc1\u660e\uff1a\u5b58\u5728\u8fb9\u957f\u662f\u6574\u6570\u4e14\u9762\u79ef\u4e3aab\u7684\u76f4\u89d2\u4e09\u89d2\u5f62\uff0e
\u8003\u70b9\uff1a\u8fdb\u884c\u7b80\u5355\u7684\u5408\u60c5\u63a8\u7406
\u4e13\u9898\uff1a\u7efc\u5408\u9898,\u63a8\u7406\u548c\u8bc1\u660e
\u5206\u6790\uff1a\u9996\u5148\u5229\u7528\u6839\u4e0e\u7cfb\u6570\u7684\u5173\u7cfb\u6c42\u5f97a+b=d-c\uff0cab=cd\uff0e\u7136\u540e\u7531\u4e09\u89d2\u5f62\u4e09\u8fb9\u5173\u7cfb\u8bc1\u5f97\u5b58\u5728\u4ee5\uff08a+b\uff09\uff0c\uff08a+c\uff09\uff0c\uff08b+c\uff09\u4e3a\u8fb9\u7684\u4e09\u89d2\u5f62\uff1b\u7531\uff08a+c\uff092+\uff08b+c\uff092=\uff08a+b\uff092\u63a8\u77e5\u8be5\u4e09\u89d2\u5f62\u662f\u76f4\u89d2\u4e09\u89d2\u5f62\uff0c\u5219\u53ef\u5f97S=ab\uff0e
\u89e3\u7b54\uff1a\u8bc1\u660e\uff1a\u7531\u9898\u8bbe\u53ef\u77e5a+b=d-c\uff0cab=cd\uff0e
\u2235a\uff0cb\uff0cc\uff0cd\u662f\u6b63\u6574\u6570\uff0c
\u2234\uff08a+b\uff09\uff0c\uff08a+c\uff09\uff0c\uff08b+c\uff09\u4efb\u610f\u4e24\u6570\u4e4b\u548c\u5927\u4e8e\u7b2c\u4e09\u4e2a\u6570\uff0c
\u4ece\u800c\u5b58\u5728\u4ee5\uff08a+b\uff09\uff0c\uff08a+c\uff09\uff0c\uff08b+c\uff09\u4e3a\u8fb9\u7684\u4e09\u89d2\u5f62\uff0e
\u2235\uff08a+c\uff092+\uff08b+c\uff092
=a2+b2+2c2+2c\uff08a+b\uff09
=a2+b2+2cd
=a2+b2+2ab
=\uff08a+b\uff092
\u2234\u8fd9\u6837\u7684\u4e09\u89d2\u5f62\u662f\u76f4\u89d2\u4e09\u89d2\u5f62\uff0c\u5176\u76f4\u89d2\u8fb9\u957f\u4e3a\uff08a+c\uff09\uff0c\uff08b+c\uff09\uff0c\u659c\u8fb9\u957f\u4e3a\uff08a+b\uff09\uff0c
\u4e14\u8be5\u4e09\u89d2\u5f62\u7684\u9762\u79ef\u4e3a\uff1aS=
1
2
\uff08a+c\uff09\uff08b+c\uff09=
1
2
[ab+c\uff08a+b+c\uff09]=
1
2
\uff08ab+cd\uff09=ab\uff0e
\u6545\u8fb9\u957f\u4e3a\uff08a+b\uff09\uff0c\uff08a+c\uff09\uff0c\uff08b+c\uff09\u7684\u4e09\u89d2\u5f62\u7b26\u5408\u9898\u8bbe\u8981\u6c42\uff0e
\u70b9\u8bc4\uff1a\u672c\u9898\u7efc\u5408\u8003\u67e5\u4e86\u6839\u4e0e\u7cfb\u6570\u5173\u7cfb\uff0c\u56e0\u5f0f\u5206\u89e3\u7684\u5e94\u7528\u4ee5\u53ca\u4e09\u89d2\u5f62\u4e09\u8fb9\u5173\u7cfb\uff0e\u6b64\u9898\u96be\u5ea6\u8f83\u5927\uff0e
①*4-②，得
4a+4b-ab-8=0
因式分解，得（a-4)(b-4)=8
∵a,b均为正整数
且8=1*8或2*4
∴a-4=1，b-4=8 或a-4=2 ，b-4=4（可互换）
所以直角三角形两直角边分别为5,12 或6,8.
所以直角三角形三边长为是5,12,13 或 6,8,10
因为a方➕b方=c方
所以c=m-2
因为a、b均为整数
所以▶️为完全平方数
x的平方-（m+2）x+4m=0
▶️=m方+12m+4
m=12
a+b=14
ab=10
a=6b=8c=10
c=m-2
M=12
a，b=6，8
c=10
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