a,

$\Delta$ NMP và $\Delta$ NHM có $\widehat{M}= \widehat{H}$, $\widehat{N}$ chung 

$\Rightarrow$ $\Delta$ NMP $\backsim$ $\Delta$ NHM (g.g)     (1)

$\Delta$ MHP và $\Delta$ NNP có $\widehat{H}= \widehat{M}= 90^o$, $\widehat{P}$ chung 

$\Rightarrow$ $\Delta$ MHP $\backsim$ $\Delta$ NMP$        (2)

(1)(2) $\Rightarrow$ $\Delta$ MHP $\backsim$ $\Delta$ NHM 

b,

$\Delta$ MNP vuông tại M có $NP= \sqrt{MN^2+MP^2}= 15cm$

(1) $\Rightarrow \frac{MH}{MN}= \frac{MP}{NP}$

$\Leftrightarrow MH= 7,2cm$

$NH= \sqrt{MN^2 - MH^2}= 5,4cm$ 

c,

MD phân giác 

$\Rightarrow$ $\frac{ND}{DP}= \frac{MN}{MP}$

$\Leftrightarrow \frac{ND}{MN}= \frac{DP}{MP}= \frac{ND+DP}{MN+MP}= \frac{15}{9+12}= \frac{5}{7}$

$\Rightarrow ND= \frac{5MN}{7}= \frac{45}{7}$

$DP= NP-ND= \frac{60}{7}$ 

d,

$\Delta$ NMH, KQ // NH có $\frac{MK}{MH}= \frac{MQ}{MN}$

$\Leftrightarrow MQ= 4,5cm$ 

Tương tự, $\Delta$ MHP có $MR= 6cm$

$\Rightarrow S_{QMR}= 0,5MQ.MR= 13,5 (cm^2)$ 

$S_{NMP}= 0,5MN.MP= 54 (cm^2)$

$\Rightarrow S_{NQRP}= 54-13,5= 40,5 (cm^2)$