the diagram that i drew is shown below:
from the diagram, you can see that the distance between building A and building B is equal to x which is equal to the distance between A and C, E and D, and F and G.
all those lines (AC, ED, FG) are equivalent to each other and can be used interchangeably in formulas involving them.
since their length is represented by x, we can use x in the equations involving their length.
from the diagram you can also see that the distance between A and E is equal to 19 is equal to the distance between C and D.
likewise, the distance between A and F is equal to 23 is equal to the distance between C and G.
lines AE and CD are equivalent and can be used interchangeably.
lines AF and CG are equivalent and can be used interchangeably.
the distance between the buildings is represented by x.
the distance between B and C is represented by y.
as far as i can tell, you have to solve for the distance between the buildings before you can solve the the height of building B.
the height of building B is represented by the line BG whose length is equal to 23 + y.
the distance from the first floor window base to the ground is not used in the calculations but it needs to be added at the end to get the height of building B.
in triangle EBD, tan(26) is equal to BD / ED
in triangle ABC, tan(6) is equal to BC / AC
since BC = y and AC = x and ED = x, this equation can be written as:
tan(26) = BD / x and tan(6) = y / x
we can solve for x in both equations to get:
x = BD / tan(26) and x = y / tan(6)
since the expressions on the right side of each equation are are both equal to each other, we can set them equal to each other to get:
BD / tan(26) = y / tan(6)
from the diagram, you can see that BD is equal to BC + CD.
you can also see that BC is equal to y.
this makes BD equal to y + CD.
CD is equal to 19.
this makes BD equal to y + 19
replace BD with y + 19 in the equation of BD / tan(26) = y / tan(6) and you get:
(y + 19) / tan(26) = y / tan(6)
cross multiply to get:
(y + 19) * tan (6) = tan(26) * y
simplify this equation to get:
y * tan(6) + 19*tan(6) = y*tan(26)
subtract y * tan(6) from both sides of this equation to get:
19*tan(6) = y*tan(26) - y*tan(6)
factor out the y on the right side of the equation to get:
19*tan(6) = y*(tan(26)-tan(6))
divide both sides of this equation by (tan(26)-tan(6)) to get:
19*tan(6) / (tan(26)-tan(6)) = y
use your calculator to get:
y = 5.21911 (rounded to 5 decimal places).
since BD is equal to y + 19, this makes BD equal to 24.21911.
you can now solve for x.
in triangle EBD, tan(26) = BD / ED
since BD is equal to 24.21911 and ED is equal to x, this equation becomes:
tan(26) = 24.21911 / x
solve for x to get:
x = 24.21911 / tan(26) which makes x = 49.65654 (rounded to 5 decimal places.
you now know the value of x and y.
x = 49.65653 which is the distance between the buildings.
y = 5.21911 which is the distance between B and C.
you can now solve for the height of building B.
the height of building B is equal to BC + 23 which makes the height of building B equal to 5.21911 + 23 = 28.21911 rounded to 5 decimal places.
you can confirm this is true by taking the tan(6) in triangle ABC and taking the tan(26) in triangle EBD.
in triangle ABC, tan(6) = BC / AC = 5.21911 / 49.65654 which is equal to .10510 rounded to 5 decimal places.
use your calculator to get tan(6) = .10510 rounded to 5 decimal places.
they match confirming the value for BC and AC are good.
note that BC = y = 5.21911
in triangle EBD, tan(26) = BD / ED = 24.21911 / 49.65654 which is equal to .48773 rounded to 5 decimal places.
use your calculator to get tan(26) = .48773 rounded to 5 decimal places.
they match confirming the value for BD and ED are good.
note that BD = BD + CD = y + 19 = 5.21911 + 19 = 24.21911.
it's a long way to the answer but i think i got it right.
check it out and see if you agree.