In $ \triangle \mathrm{ABC}, \angle \mathrm{B}=90^{\circ} $ and $ \mathrm{BM} $ is an altitude. If $ \mathrm{AM}=2 x^{2} $ and $ \mathrm{CM}=8 x^{2} $, find $ \mathrm{BM} $, AB and $ \mathrm{BC} $.

Given:

In \( \triangle \mathrm{ABC}, \angle \mathrm{B}=90^{\circ} \) and \( \mathrm{BM} \) is an altitude.

\( \mathrm{AM}=2 x^{2} \) and \( \mathrm{CM}=8 x^{2} \)

To do:

We have to find \( \mathrm{BM} \), AB and \( \mathrm{BC} \).

Solution:

In $\triangle ABC$,

By Pythagoras theorem,

$AC^2 =AB^2+BC^2$

$(2x^2+8x^2)^2=AB^2+BC^2$

$(2x^2)^2+(8x^2)^2+2\times2x^2\times8x^2=AB^2+BC^2$

$(2x^2)^2+(8x^2)^2+32x^4=AB^2+BC^2$.........(i)

Similarly,

In $\triangle ABM$,

By Pythagoras theorem,

$AB^2 =AM^2+BM^2$

$AB^2=(2x^2)^2+BM^2$......(ii) 

In $\triangle BMC$,

By Pythagoras theorem,

$BC^2 =MC^2+BM^2$

$BC^2=(8x^2)^2+BM^2$......(iii)

From (i), (ii) and (iii),

$(2x^2)^2+(8x^2)^2+32x^4=(2x^2)^2+BM^2+(8x^2)^2+BM^2$

$32x^4=2BM^2$

$16x^4=BM^2$

$BM^2=(4x^2)^2$

$BM=4x^2$

Therefore,

$AB^2=(2x^2)^2+16x^4$

$=4x^4+16x^4$

$=20x^4$

$\Rightarrow AB=\sqrt{20x^4}$

$=2\sqrt5x^2$

$BC^2=(8x^2)^2+16x^4$

$=64x^4+16x^4$

$=80x^4$

$\Rightarrow BC=\sqrt{80x^4}$

$=4\sqrt5x^2$