Part a) R(x-axis) is the reflection of the original triangle ABC on the x-axis. The new coordinates are given as A' (2, -5), B' (4, -6), and C' (3, -1)
Part b) R(y=3) is the reflection of the original triangle ABC on the line with equation y=3. The new coordinates would be A' (2, 1), B' (4, 0), and C' (3, 5)
Part c) T(-2, 5) is the translation of the original triangle ABC two units left and five units up. The new coordinates would be A'(0, 10), B' (2, 11), and C'(1, 6)
Part d) T(3, -6) is the translation of the original triangle ABC three units right and six units down. The new coordinates would be A'(5, -1), B'(7, 0), and C'(6, -5)
Part e) r(90°, 0) is the rotation of the original triangle ABC on the origin by 90° clockwise. The new coordinates would be A'(5, -2), B'(6, -4) and C'(1 -3)
The two pairs are Pythagorean triples because if you plug the two legs of a right triangle into the Pythagorean theorem(A^2+B^2=C^2), then you will find the measurement for the third side(hypotenuse). i.e. 15^2+12^2=9^2(it's the Pythagorean triple 3,4,5 multiplied by 3). This works with any triple, as long as your using the legs of the triangle and as long as the triangle is a right triangle.