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24. (9分)【阅读】若$x$满足$(10 - x)(x - 3)=17$,求$(10 - x)^2 + (x - 3)^2$的值.
设$10 - x = a$,$x - 3 = b$,
则$(10 - x)(x - 3)=ab = 17$,$a + b=(10 - x)+(x - 3)=7$,$(10 - x)^2 + (x - 3)^2=a^2 + b^2=(a + b)^2 - 2ab=7^2 - 2×17=15$.
(1)【理解】
①若$x$满足$(50 - x)(x - 35)=100$,则$(50 - x)^2 + (x - 35)^2$的值为________;
②若$x$满足$(x - 1)(3x - 7)=\frac{7}{6}$,试求$(7 - 3x)^2 + 9(x - 1)^2$的值;
(2)【应用】
如图,在长方形$ABCD$中,$AD = 2CD = 2x$,$AE = 44$,$CG = 30$,长方形$EFGD$的面积是$200$,四边形$NGDH$和$MEDQ$都是正方形,四边形$PQDH$是长方形,延长$MP$至点$T$,使$PT = PQ$,延长$MF$至点$O$,使$FO = FE$,过点$O$,$T$作$MO$,$MT$的垂线,两垂线相交于点$R$,求四边形$MORT$的面积.
(结果必须是一个具体的数值)
设$10 - x = a$,$x - 3 = b$,
则$(10 - x)(x - 3)=ab = 17$,$a + b=(10 - x)+(x - 3)=7$,$(10 - x)^2 + (x - 3)^2=a^2 + b^2=(a + b)^2 - 2ab=7^2 - 2×17=15$.
(1)【理解】
①若$x$满足$(50 - x)(x - 35)=100$,则$(50 - x)^2 + (x - 35)^2$的值为________;
②若$x$满足$(x - 1)(3x - 7)=\frac{7}{6}$,试求$(7 - 3x)^2 + 9(x - 1)^2$的值;
(2)【应用】
如图,在长方形$ABCD$中,$AD = 2CD = 2x$,$AE = 44$,$CG = 30$,长方形$EFGD$的面积是$200$,四边形$NGDH$和$MEDQ$都是正方形,四边形$PQDH$是长方形,延长$MP$至点$T$,使$PT = PQ$,延长$MF$至点$O$,使$FO = FE$,过点$O$,$T$作$MO$,$MT$的垂线,两垂线相交于点$R$,求四边形$MORT$的面积.
(结果必须是一个具体的数值)
答案
【点拨】本题考查完全平方公式,熟练掌握完全平方公式是解题的关键.
【解析】(1)①令$a = 50 - x$,$b = x - 35$,因为$(50 - x)(x - 35) = 100$,所以$ab = 100$,$a + b = 15$,所以$(a + b)^2 = a^2 + b^2 + 2ab = 225$,所以$a^2 + b^2 = 225 - 2ab = 25$,所以$(50 - x)^2 + (x - 35)^2 = a^2 + b^2 = 25$. 故答案为 25.
②因为$(x - 1)(3x - 7) = \dfrac{7}{6}$,所以$3(x - 1)(7 - 3x) = -\dfrac{7}{6}×3 = -\dfrac{7}{2}$,令$a = 3(x - 1)$,$b = 7 - 3x$,所以$ab = -\dfrac{7}{2}$,$a + b = 4$,所以$(a + b)^2 = a^2 + b^2 + 2ab = 16$,所以$a^2 + b^2 = 16 - 2ab = 23$,所以$(7 - 3x)^2 + 9(x - 1)^2 = b^2 + a^2 = 23$.
(2)因为$ED = AD - AE$,$DG = DC - CG$,所以$ED = 2x - 44$,$DG = x - 30$,所以$MT = MO = (2x - 44) + 2(x - 30)$. 因为长方形$EFGD$的面积是 200,所以$(2x - 44)(x - 30) = 200$,所以$2(x - 30)(2x - 44) = 400$. 令$a = 2x - 44$,$b = 2(x - 30)$,所以$ab = 400$,$a - b = 16$,所以$(a - b)^2 = a^2 + b^2 - 2ab = 256$,所以$a^2 + b^2 = 256 + 2ab = 1\ 056$,所以四边形$MORT$的面积$= MT^2 = (a + b)^2 = a^2 + b^2 + 2ab = 1\ 056 + 800 = 1\ 856$.
【解析】(1)①令$a = 50 - x$,$b = x - 35$,因为$(50 - x)(x - 35) = 100$,所以$ab = 100$,$a + b = 15$,所以$(a + b)^2 = a^2 + b^2 + 2ab = 225$,所以$a^2 + b^2 = 225 - 2ab = 25$,所以$(50 - x)^2 + (x - 35)^2 = a^2 + b^2 = 25$. 故答案为 25.
②因为$(x - 1)(3x - 7) = \dfrac{7}{6}$,所以$3(x - 1)(7 - 3x) = -\dfrac{7}{6}×3 = -\dfrac{7}{2}$,令$a = 3(x - 1)$,$b = 7 - 3x$,所以$ab = -\dfrac{7}{2}$,$a + b = 4$,所以$(a + b)^2 = a^2 + b^2 + 2ab = 16$,所以$a^2 + b^2 = 16 - 2ab = 23$,所以$(7 - 3x)^2 + 9(x - 1)^2 = b^2 + a^2 = 23$.
(2)因为$ED = AD - AE$,$DG = DC - CG$,所以$ED = 2x - 44$,$DG = x - 30$,所以$MT = MO = (2x - 44) + 2(x - 30)$. 因为长方形$EFGD$的面积是 200,所以$(2x - 44)(x - 30) = 200$,所以$2(x - 30)(2x - 44) = 400$. 令$a = 2x - 44$,$b = 2(x - 30)$,所以$ab = 400$,$a - b = 16$,所以$(a - b)^2 = a^2 + b^2 - 2ab = 256$,所以$a^2 + b^2 = 256 + 2ab = 1\ 056$,所以四边形$MORT$的面积$= MT^2 = (a + b)^2 = a^2 + b^2 + 2ab = 1\ 056 + 800 = 1\ 856$.
