Question

Find the diagonal of a cube if its side equals 5. When applicable, simplify radicals and show all of your work. Help plz

Answer 1

1. Hi, please take a good look at the pictures at the bottom.

2. What we are trying to find here is BH. 

3. The second picture shows the "diagonal section", DBFH in another perspective. 

4. We notice that HB is the hypothenuse of right triangle DBH.

5. So we can apply the pythegoran theorem to find HB, but first we need to find HD and DB.

6. HD is just a side of the cube so it is 5 units. DB is the diagonal of the base square  ABCD, so we find it by applying the pythagorean theorem in right triangle ABD: 
$AB^2+AD^2=BD^2$

$5^2+5^2=BD^2 2*5^2=BD^2 \sqrt{2*5^2} = \sqrt{BD^2} BD=5 \sqrt{2}$

6. So we have both HD and DB:

$HD^2+DB^2=HB^2 5^2+ (5 \sqrt{2} )^{2}=HB^2 25+50=HB^2 HB^2=75 HB= \sqrt{25*3}= \sqrt{25} * \sqrt{3}= 5\sqrt{3}$

7. Remark: Another way would be to use the generalized Pythagorean theorem, which is as follows:

$Diagonal^2=5^2+5^2+5^2 Diagonal = \sqrt{3* 5^{2} } =5 \sqrt{3}$


8. In general, given a right rectangular prism with sides a, b and c, the diagonal is given by $\sqrt{a^2+b^2+c^2}$