Giải thích các bước giải:
a.Ta có $\Delta MNP$ vuông tại $M$
$\to NP=\sqrt{MN^2+MP^2}=10$
Mà $MH\perp NP$
$\to MH\cdot NP=MN\cdot MP(=2S_{MNP})$
$\to MH=\dfrac{MN\cdot MP}{NP}=\dfrac{24}5$
b.Xét $\Delta MPH,\Delta MNP$ có:
Chung $\hat P$
$\widehat{MHP}=\widehat{NMP}(=90^o)$
$\to\Delta PHM\sim\Delta PMN(g.g)$
$\to\dfrac{PH}{PM}=\dfrac{PM}{PN}$
$\to PM^2=PH\cdot PN$
c.Từ câu b
$\to\dfrac{S_{MHP}}{S_{MNP}}=(\dfrac{PM}{PN})^2=\dfrac{16}{25}$
$\to S_{MHP}=\dfrac{16}{25}S_{MNP}=32$ vì $S_{MNP}=50$
d.Ta có $MD, NE, PF$ là phân giác $\Delta MNP$
$\to\dfrac{ME}{EP}=\dfrac{MN}{NP}, \dfrac{PD}{DN}=\dfrac{MP}{MN}, \dfrac{NF}{FM}=\dfrac{PN}{PM}$
$\to \dfrac{ME}{EP}\cdot\dfrac{PD}{DN}\cdot\dfrac{NF}{FM}=\dfrac{MN}{NP}\cdot\dfrac{MP}{MN}\cdot\dfrac{PN}{PM}=1$