Hmm, I used a different method and overlooking my previous answer have to change the number. I made a mistake in my previous calculation. Hope I got it right this time. Math for amateurs, wasn't it? :-)
I used the diagonal of the square 60/60. The length of that diagonal is 60√2.
Then I made use of the sinus-rule which says that:
A/sin α = B/sin β = C/ sin γ.
Then I reasoned that there were equal angles in both triangles divided by this diagonal.
The angle in the lower right corner beneath the diagonal is equal to the top angle of the big triangle and the angle in the right corner above the diagonal is equal to the left below angle of the big triangle. All those angles must be 45°, otherwise will you not get the square of 60 by 60.
Then can you apply the above mentioned sinus rule again and do you get the next calculation:
x/ sin 45° = 60√2 /45° and (220-x)/ sin 45° = 60√2 /45°. It is clear that x equals 60√2.
With the help of Pythagoras does that imply that the lower bottom of the triangle is 120 wide.
The standing height is therefor:
√(220² – 120²) = 184.
That is different then my first answer, because I made an error in that calculation. I took wrong numbers. I had made several attempts and calculated using wrong numbers (110 as the value for x).
I used slope to find it
.
Assuming the intersection of the H side and the ground is the origin, the equation of the ladder is: y-60 = m(x+60)
Then: y = mx + 60m + 60.
It follows that the y intercept (H) is 60m + 60, and the x intercept is -60-60/m.
So (60m+60)^2 + (-60-60/m)^2 = 220^2 by pythagorean theorum.
My calculator solved this for 4 different values, but because the slope has to clearly be greater than 1, the only one that worked was m = 2.3805.
Plugging this into the y intercept of H = 60m+60, I got 202.83.
Didn't use any similar triangles. I'm not really a geometry person though, as I stated earlier.