Streak Geometry.

Area of pllgm- base ht=15 30=450

Area of tri-=450 (given)

Base of triangle=0.75*30=22.5

Area of triangle- 0.5 b h= 0.5 22.5 h= 450

h=40

$30 cm \times 15 cm = 450 cm^{2}$

$\rightarrow$ Let the height of triangle to $h$ .

$\Rightarrow 30 \times \frac{3}{4} \times h \times \frac{1}{2} = 450$

$30h \times \frac{3}{8} = 450$

$30h \times 3 = 3600$

$90h = 3600$

$\therefore h = 40$

Therefore, the height of the triangle is $\boxed{40 cm}$ .

Let $h$ be the height of the new triangle. We have that $30\times 15=\dfrac{(30\times 0.75)\times h}{2}$

Note that the $30$ 's cancel out, giving $15=\dfrac{0.75h}{2}$

$30=\dfrac{3}{4}h$

$h=\boxed{40}$

I'll present a wordy solution:

For the sake of simplicity, first imagine the target figure to be a parallelogram instead of a triangle.

The base is 3/4 th of the old base. So, we must make the new height 4/3 times the old height in order to keep the area same.

But again, the target figure is actually a triangle, not a parallelogram. Since the area of the triangle is 1/2 of the parallelogram, we must multiply the area with 2 to cope up.

Thus the new height is 8/3 times the old height.

Hence, the answer is 40 cm.

An esteemed reader might choose to ask me why I wrote a wordy solution rather than some simple algebra. My answer is that I wrote it this way because this is how I did it in my head as I had no pencil and paper :/