Question

PLEASE HELP

What are the values of a and b?

a = 28, b = 3√37
a = 324, b = 333
a = 18, b = 3√37
a = 18, b = 18√37

Answer 1

In the smallest of the three triangles (with sides
`a,b,3`) we have

`a^2+3^2=b^2`

In the second-largest triangle (with sides
`a,108`), the length of the missing side is
`√(a^2+108^2)`. Then in the largest triangle (with sides
`b,111,√(a^2+108^2)`), we have

`b^2+\left(√(a^2+108^2)\right)^2=(108+3)^2=111^2`

`\implies b^2+a^2+108^2=111^2`

`\implies b^2+a^2=657`

Equivalently, we have

`(a^2+9)+a^2=657\implies 2a^2=648\implies a^2=324\implies a=18`

In turn, we find

`b^2=18^2+3^2=298\implies b=√(333)=3√(37)`