Question

Triangle A has sides of lengths `36`, `42`, and `48`. Triangle B is similar to triangle A and has a side of length `12`. What are the possible lengths of the other two sides of triangle B?

Answer 1

Other two sides of `B`:
`color(white)("XXX"){14,16}` or
`color(white)("XXX"){ 10 2/7, 13 3/7}` or
`color(white)("XXX"){9, 10 1/2}`

Explanation:

Option 1: B's side with length `color(blue)(12)` corresponds to A's side with length `color(blue)(36)`
Ratio lengths `B:A = 12:36 = 1/3`
#{: ("A's side",rarr,"B's side"), (36,rarr,1/3 * 36=12), (42,rarr,1/3 * 42=14), (48,rarr,1/3 * 48 = 16) :}#

Option 2: B's side with length `color(blue)(12)` corresponds to A's side with length `color(blue)(42)`
Ratio lengths `B:A = 12:42 = 2/7`
#{: ("A's side",rarr,"B's side"), (36,rarr,2/7 * 36=10 2/7), (42,rarr,2/7 * 42=12), (48,rarr,2/7 * 48 = 13 3/7) :}#

Option 1: B's side with length `color(blue)(12)` corresponds to A's side with length `color(blue)(48)`
Ratio lengths `B:A = 12:48 = 1/4`
#{: ("A's side",rarr,"B's side"), (36,rarr,1/4 * 36=9), (42,rarr,1/4 * 42=10 1/2), (48,rarr,1/4 * 48 = 12) :}#