I've modified my first statement and I hope it is now better understandable what I meant:

1. All slanted sides are isosceles triangles. Heights and medians of these triangles are the same. (blue) Use Pythagorean theorem to calculate it's length.

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Thank you so much for your reply. Isn't the slant height what you marked as h? **<<<<<< correct**

When I did as you said, I looked at the answer posted, and it was different. I got a height of around 7, but the answer says 22.48. Can you show how to get this? Again, thanks.

1. Since the base lines of the isosceles triangles are all different the slant heights in these triangles must be different too.

2. I've attached a completed sketch which I use for the following calculations:

\(\displaystyle h=\sqrt{24^2-12.5^2}=\sqrt{\frac{1679}4}\approx 20.4878\)

\(\displaystyle \cos(S)=\dfrac{50^2-39^2-25^2}{-2 \cdot 39 \cdot 25}~\implies~S\approx 100.4594^\circ\)

\(\displaystyle m=\sqrt{39^2+12.5^2-2\cdot 12.5 \cdot 39 \cdot \cos(S)}=\sqrt{1854.25}\approx 43.0610\)

\(\displaystyle \cos(T)=\dfrac{h^2-m^2-24^2}{-2 \cdot m \cdot 24}~\implies~T\approx 13.4189^\circ\)

\(\displaystyle \boxed{H=24 \cdot \sin(T)\approx 5.5696}\)