Ta có: \(\hat{MTP}+\hat{MTN}=180^0\) (hai góc kề bù)
=>\(\hat{MTN}=180^0-120^0=60^0\)
Xét ΔMTN có \(cosMTN=\frac{TM^2+TN^2-MN^2}{2\cdot TM\cdot TN}\)
=>\(\frac{4^2+TN^2-5^2}{2\cdot4\cdot TN}=cos60=\frac12\)
=>\(TN^2-9=\frac12\cdot8\cdot TN=4\cdot TN\)
=>\(TN^2-4\cdot TN-9=0\)
=>\(TN^2-4\cdot TN+4-13=0\)
=>\(\left(TN-2\right)^2=13\)
=>\(\left[\begin{array}{l}TN-2=\sqrt{13}\\ TN-2=-\sqrt{13}\end{array}\right.\Rightarrow\left[\begin{array}{l}TN=2+\sqrt{13}\left(nhận\right)\\ T=-\sqrt{13}+2\left(loại\right)\end{array}\right.\)
=>\(TN=2+\sqrt{13}\)
Xét ΔMNP có MT là phân giác
nên \(\frac{PT}{PM}=\frac{NT}{NM}=\frac{2+\sqrt{13}}{5}\)
=>\(\frac{PT}{2+\sqrt{13}}=\frac{PM}{5}\)
Đặt \(\frac{PT}{2+\sqrt{13}}=\frac{PM}{5}=k\) (Điều kiện: k>0)
=>\(PT=k\left(2+\sqrt{13}\right);PM=5k\)
Xét ΔTMP có \(cosTMP=\frac{TP^2+TM^2-MP^2}{2\cdot TM\cdot TP}\)
=>\(cos120=\frac{TP^2+4^2-MP^2}{2\cdot4\cdot TP}=\frac{TP^2-MP^2+16}{8\cdot TP}\)
=>\(TP^2-MP^2+16=8\cdot TP\cdot\frac{-1}{2}=-4\cdot TP\)
=>\(k^2\left(2+\sqrt{13}\right)^2-\left(5k\right)^2+16=-4k\left(2+\sqrt{13}\right)\)
=>\(k^2\left(17+4\sqrt{13}-25\right)+16+4k\left(2+\sqrt{13}\right)=0\)
=>\(k^2\left(4\sqrt{13}-8\right)+4k\left(2+\sqrt{13}\right)+16=0\)
=>\(k^2\left(\sqrt{13}-2\right)+k\left(2+\sqrt{13}\right)+4=0\) (1)
\(\Delta=\left(2+\sqrt{13}\right)^2-4\left(\sqrt{13}-2\right)\cdot4=17+4\sqrt{13}-16\left(\sqrt{13}-2\right)=17+4\sqrt{13}-16\sqrt{13}+32=49-12\sqrt{13}=\left(6-\sqrt{13}\right)^2\)
=>(1) có hai nghiệm phân biệt là:
\(\left[\begin{array}{l}k=\frac{-\left(2+\sqrt{13}\right)-\sqrt{\left(6-\sqrt{13}\right)^2}}{2\left(\sqrt{13}-2\right)}=\frac{-2-\sqrt{13}-6+\sqrt{13}}{2\left(\sqrt{13}-2\right)}=-\frac{8}{2\left(\sqrt{13}-2\right)}=\frac{4}{2-\sqrt{13}}<0\left(loại\right)\\ k=\frac{-\left(2+\sqrt{13}\right)+\sqrt{\left(6-\sqrt{13}\right)^2}}{2\left(\sqrt{13}-2\right)}=\frac{-2-\sqrt{13}+6-\sqrt{13}}{2\left(\sqrt{13}-2\right)}=\frac{4-2\sqrt{13}}{-2\left(2-\sqrt{13}\right)}=\frac{2\left(2-\sqrt{13}\right)}{-2\left(2-\sqrt{13}\right)}=-1\left(loại\right)\end{array}\right.\)
=>Không tồn tại số đo cạnh MP thỏa mãn đề bài
