2024 AMC 12B Problem 21

The measures of the smallest angles of three different right triangles sum to 90^\circ. All three triangles have side lengths that are primitive Pythagorean triples. Two of them are 3-4-5 and 5-12-13. What is the perimeter of the third triangle?

\textbf{(A) }40 \qquad \textbf{(B) }126 \qquad \textbf{(C) }154 \qquad \textbf{(D) }176 \qquad \textbf{(E) }208 \qquad

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.