Find a primitive Pythagorean triple (a, b, c) with b = 4T5. Do the same for b = 4T6 and for b = 4T7. (b) Do you think that for every triangular number T n, there is a primitive Pythagorean triple (a, b, c) with b = 4Tn? If you believe that this is true, then prove it. Otherwise, f ind some triangular number for which it is not true.

The diagram shown in Figure 1 was used by the Hindu mathematician Bhaskara to prove the theorem in the 12th century. His proof consisted only of the diagram and the word Behold! Use this figure to derive the Pythagorean Theorem. The right triangle with sides a, b, and c has been repeated three times. The area of the large square is equal to the sum of the areas of the four triangles and the area of the small square in the center. The key to this derivation is in finding the length of a side of the small square. Once you have the equation, simplify the right side to obtain the theorem. Figure 1