若三角形的三边长是a,b,c,且满足a²+2b²+c²-2ab-2bc=0,试判断三角形的形状.小明是这样做的:解:∵a²+2b²+c²-2ab-2bc=0,∴(a²-2ab+b²)+(b²-2bc+c²)=0即(a-b)²+(b-c)²
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![若三角形的三边长是a,b,c,且满足a²+2b²+c²-2ab-2bc=0,试判断三角形的形状.小明是这样做的:解:∵a²+2b²+c²-2ab-2bc=0,∴(a²-2ab+b²)+(b²-2bc+c²)=0即(a-b)²+(b-c)²](/uploads/image/z/10151753-41-3.jpg?t=%E8%8B%A5%E4%B8%89%E8%A7%92%E5%BD%A2%E7%9A%84%E4%B8%89%E8%BE%B9%E9%95%BF%E6%98%AFa%2Cb%2Cc%2C%E4%B8%94%E6%BB%A1%E8%B6%B3a%26sup2%3B%2B2b%26sup2%3B%2Bc%26sup2%3B-2ab-2bc%3D0%2C%E8%AF%95%E5%88%A4%E6%96%AD%E4%B8%89%E8%A7%92%E5%BD%A2%E7%9A%84%E5%BD%A2%E7%8A%B6.%E5%B0%8F%E6%98%8E%E6%98%AF%E8%BF%99%E6%A0%B7%E5%81%9A%E7%9A%84%EF%BC%9A%E8%A7%A3%EF%BC%9A%E2%88%B5a%26sup2%3B%2B2b%26sup2%3B%2Bc%26sup2%3B-2ab-2bc%3D0%2C%E2%88%B4%28a%26sup2%3B-2ab%2Bb%26sup2%3B%29%2B%28b%26sup2%3B-2bc%2Bc%26sup2%3B%29%3D0%E5%8D%B3%EF%BC%88a-b%29%26sup2%3B%2B%EF%BC%88b-c%29%26sup2)
若三角形的三边长是a,b,c,且满足a²+2b²+c²-2ab-2bc=0,试判断三角形的形状.小明是这样做的:解:∵a²+2b²+c²-2ab-2bc=0,∴(a²-2ab+b²)+(b²-2bc+c²)=0即(a-b)²+(b-c)²
若三角形的三边长是a,b,c,且满足a²+2b²+c²-2ab-2bc=0,试判断三角形的形状.小明是这样做的:
解:∵a²+2b²+c²-2ab-2bc=0,∴(a²-2ab+b²)+(b²-2bc+c²)=0
即(a-b)²+(b-c)²=0.
∵(a-b)²≥0,(b-c)²≥0,∴a=b,b=c,即a=b=c.
∴该三角形是等边三角形.
仿照小明的解法解答问题:
已知:a,b,c,为三角形的三条边,且a²+b²+c²-ab-bc-ac=0,试判断三角形的形状
若三角形的三边长是a,b,c,且满足a²+2b²+c²-2ab-2bc=0,试判断三角形的形状.小明是这样做的:解:∵a²+2b²+c²-2ab-2bc=0,∴(a²-2ab+b²)+(b²-2bc+c²)=0即(a-b)²+(b-c)²
a²+b²+c²-ab-bc-ac=0得到
2a²+2b²+2c²-2ab-2bc-2ac=0
(a-b)^2+(a-c)^2+(b-c)^2=0 平方为非负数,它们的和为0,只有分别等于0
即a-b=0,a-c=0,b-c=0
a=b=c
等边三角形
asaks
∵a² + b² + c² - ab - bc - ac = 0
∴2a² + 2b² + 2c² - 2ab - 2bc - 2ac = 0
∴(a² - 2ab + b²) + (b² - 2bc + c²) + (a² - 2ac + c²) = 0
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∵a² + b² + c² - ab - bc - ac = 0
∴2a² + 2b² + 2c² - 2ab - 2bc - 2ac = 0
∴(a² - 2ab + b²) + (b² - 2bc + c²) + (a² - 2ac + c²) = 0
∴(a - b)² + (b - c)² + (a - c)² = 0
∵(a - b)²≥ 0,(b - c)²≥0,(a - c)²≥0,
∴a = b , b = c, a = c ,即 a = b = c
∴该三角形是等边三角形。
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