How to find the height of a tilted rectangle

Question

Suppose we are given two congruent rectangles ABCD and EFGH as shown in the figure, with AB = 8 , AD = 4 and EA = 3.

Find the distance of point G from line AB; that is, find X.

Answer 1

You do not need to compute angles and doing that will lead to rounding errors, besides wasting time. Can be done with three calculations (two multiplications and one addition); not worth getting out the calculator.

Draw a vertical line through F meeting the bottom horizontal line AB at T and the top horizontal line through G (extended) at S.

Triangle FTE is similar to triangle DAE

FT/FE = DA/DE

DE = 5 by Pythagoras 3-4-5 triangle

FT/8 = 4/5

FT = 32/5 or 6.4

Triangle GSF is also similar to triangle DAE (work out parallel lines and right angles and you will see this has to be true)

(.Please note: triangle GSF is positioned "sideways" in comparison to triangle DAE. Work out the equal angles with right angles and parallel lines given, and note corresponding lines carefully)

FS/FG = EA/ED

FS/4 = 3/5

FS = 12/5 or 2.4

X = ST = FS + FT = 32/5 + 12/5 = 44/5 or 8.8 exactly.

Double-check my calculations, always, but method is good.

The top diagram added below is helpful; thank you for the assistance.

PLEASE NOTE: The diagram kindly added below has a small misinterpretation included (as of 2:20PM EST; hopefully it will be corrected). FS is supposed to be vertical, not horizontal; FS is an extension of TF. S is supposed to be on the horizontal line through G shown in the original diagram, so that ST = X, the height to be found. This present diagram is not consistent with the description and solution above. Please make the correction, thank you.

Answer 2

Step 1. Calculate the angle of the diagonal AC.

Quickest way is to get the arctan(4/8) which gives 26.57 degrees.

Step 2. Calculate the length of the diagonal AC. Use pythagoras. (you should get 8.94)

Step 3. Calculate the angle of rotation of EF.

Again, quickest way is to get the arctan(4/3) which gives 53.13 degrees.

Therefore the total rotation of the diagonal is 26.57 degrees + 53.17 degrees. The length of the diagonal calculated in step 2 times the sine of the total rotation gives the height X you are looking for (you should get 8.8)

Answer 3

Diagonal length of block $ AC = EG = 4 \sqrt 5,$ which is hinged up through $ \sin^{-1} \frac{4}{5}.$

That makes a raise of $ 4 \sqrt 5 \cdot \dfrac{4}{5} = \dfrac{16}{ \sqrt 5 }\approx 7.15542 $