triangular number building blocks

(1) it is easy to show that a triangular number plus the next triangular number make a square number

(2) can students show that 8 of a triangular number plus 1 makes a square number? (a result credited to Plutarch (100CE))

the algebraic justification for this is reasonably straightforward
a diagram is probably more (immediately) compelling:

(3) can students show that a triangular number plus 6 times the next triangular number plus the next triangular number is a square number?
using a diagram?
using algebra?

(4) can students show with a diagram that 9 of a triangular number plus 1 makes another triangular number?

the algebraic justification for this is a little more demanding
however, there is a neat associated diagram: