consider the right traingel with sices 3, 4, and 5. the lengths form an arithmetic series.

a. what are the other three right triangles in arthimetic series

Actually, I think there are an infinite numbr of such triangles.

You want to have sides a, b and c such that a < b < c and b - a = c - b.

This means that:

c = 2 b - a

You also know that:

c^2 = a^2 + b^2 --->

c^2 - b^2 = a^2

(c-b)(c+b) = a^2

(b-a)(3b-a) = a^2 --->

3 b^2 - 4ba = 0 --->

3b - 4a = 0 --->

b = 4/3 a

E.g.

take a = 27, then b = 36 and c should be 36 plus the difference of 36 and 27, which is 45. Pythagoras gives
c = sqrt[27^2 + 36^2] which is indeed 45.

To find the other three right triangles in an arithmetic series similar to the given triangle with sides 3, 4, and 5, you can use the following steps:

1. Define the difference (d) between the sides of the triangles. Since it is an arithmetic series, the difference will be constant.

2. Set up the relation between the sides using the variables a, b, and c. We know that a < b < c, and b - a = c - b.

3. Using the relation c = 2b - a, you can express the larger side (c) in terms of the other two sides (a and b).

4. Apply the Pythagorean theorem: c^2 = a^2 + b^2. This equation will help us solve for the other sides.

5. Simplify the equation to (b - a)(3b - a) = a^2.

6. Use the equation 3b - 4a = 0, derived from simplifying the equation above, to determine the ratio between b and a.

7. Choose a value for a, which represents the smaller side of the first triangle in the series.

8. Calculate b using the ratio b = 4/3 * a.

9. Find c by substituting the values of a and b into the equation c = 2b - a.

10. Use the Pythagorean theorem to confirm that c is the hypotenuse and satisfies the equation c^2 = a^2 + b^2.

By following these steps, you can generate three additional right triangles in an arithmetic series similar to the given triangle. Remember, there are an infinite number of such triangles since the difference can take any value.