Question

Triangle A has sides of lengths `32`, `36`, and `24`. Triangle B is similar to triangle A and has a side of length `8`. What are the possible lengths of the other two sides of triangle B?

Answer 1

The other two sides of B are
`color(white)("XXX"){9,6}`
or
`color(white)("XXX"){7 1/9, 5 1/3}`
or
`color(white)("XXX"){12,10 2/3}`

Explanation:

Denote the given side of B as `B_1=8`
and
the sides of A as `A_1, A_2, A_3` such that `A_1` corresponds to `B_1`

Given: `{A_1,A_2,A_3}={32,36,24}`

Case 1 : `A_1=32`
`color(white)("XXX")`The ratio of `A_1:B_1 = 32:8=4:1`
`color(white)("XXX")`All corresponding sides must be in this ratio
`color(white)("XXX")rarr B_x=A_x/4`
`color(white)("XXX")`So the other two sides of B must have lengths
`color(white)("XXX")36/4=9 and 24/4=6`

Case 1 : `A_1=36`
`color(white)("XXX")`The ratio of `A_1:B_1 = 36:8=4.5:1`
`color(white)("XXX")`All corresponding sides must be in this ratio
`color(white)("XXX")rarr B_x=A_x/4.5`
`color(white)("XXX")`So the other two sides of B must have lengths
`color(white)("XXX")32/4.5=7 1/9 and 24/4.5=5 1/3`

Case 1 : `A_1=24`
`color(white)("XXX")`The ratio of `A_1:B_1 = 24:8=3`
`color(white)("XXX")`All corresponding sides must be in this ratio
`color(white)("XXX")rarr B_x=A_x/3`
`color(white)("XXX")`So the other two sides of B must have lengths
`color(white)("XXX")36/3=12 and 32/3=10 2/3`