he hypotenuse of a right triangle is two more than the longer leg, and the longer leg is 47 more than the shorter leg. Find all three sides of the triangle.

Start by expressing all the legs in terms of one variable. Since you know the relationship of both the hypotenuse and the shorter leg to the longer leg, you can use the longer leg to express the length of the other two legs.

Let's call the length of the longer leg, a. We know that the hypotenuse, let's call it c, is two more than the longer leg. That means c = a + 2. We also know the longer leg is 47 more than the shorter leg. Let's call the shorter leg b. That means b = a - 47. See picture.

Now we know a, b, and c all in terms of one variable, a. Remember that the Pythagorean Theorem states that [tex] a^{2} + b^{2} = c^{2} [/tex], where a and b = length of the legs and c = length of the hypotenuse. Plug your values for a, b, and c into the Pythagorean Theorem and solve for a:
[tex]a^{2} + b^{2} = c^{2} \\ a^{2} + (a - 47)^{2} = (a + 2)^{2} \\ a^{2} + a^{2} - 94a + 2209 = a^{2} + 4a + 4\\ a^{2} - 98a + 2205 = 0\\ (a - 35)(a - 63) = 0\\ a = 35, a = 63[/tex]

Now you have two values for a: 35 and 63. However, remember that the length of your shorter leg, b, equals a - 47. The length must be a positive number, that means a must equal 63. 

Since you know that a = 63, plug that into your equations for the lengths of each side to get the sides of the triangle:
a (longer leg) = 63
b (shorter leg) = a - 47 = 63 - 47 = 16
c (hypotenuse) = a + 2 = 63 + 2 = 65

Your three sides are 63, 16, and 65.