Lngth of a diagonal

Question

The application fo the Pythagorean Theorem in three dimensions involves the relationship betwee teh perpendicular edges of a recangular block and the solid diagonal of the same block.

In the figure, show that h^2=a^2+b^2=c^2

 

The figure is a rectangular prism. The diagonal from the lower corner to the top corner is h. The base sides are a and b. The height is c. 

Answer 1

x is the horizontal axis (east to west)

y is the vertical axis (north to south)

z is the 3-D axis ( into the screen and towards you)

3-D pythagorean formula is h^2 = sqrt( x^2 + y^2 + z^2)

and the 3-D distance formula is

D = sqrt ( (x1-x2)^2 + (y1-y2)^2 + (z1 - z2)^2)

Ex. (12,8,5) vs ( 3,2,3)

the distance is sqrt ( (12-3)^2 + (8-2)^2 + (5-3)^2)

= sqrt ( 9^2 + 6^2 + 2^2) =

sqrt ( 81 + 36 + 4) =

sqrt( 121) = 11