The three sides of a right triangle form an arithmetic sequence. Which of the following must be true about the length of the shortest side?ResponsesIt will be a multiple of 3.It will be a multiple of 3.It will equal three times the common difference.It will equal three times the common difference.It will equal three more than the common difference.It will equal three more than the common difference.None of the above must be true

The correct answer is "It will equal three times the common difference."

Here's why:

In an arithmetic sequence, the difference between any two consecutive terms is constant. This constant difference is known as the common difference.

If the three sides of a right triangle form an arithmetic sequence, let's denote the sides as a, a+d, a+2d, where a is the shortest side, d is the common difference, and a+2d is the hypotenuse (the longest side).

According to the Pythagorean theorem, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. So, we have:

(a+2d)^2 = a^2 + (a+d)^2

Expanding and simplifying this equation, we get:

a^2 + 4ad + 4d^2 = a^2 + 2ad + d^2 + a^2 + 2ad + d^2

This simplifies to:

4ad + 4d^2 = 2a^2 + 4ad + 2d^2

Subtracting 4ad and 2d^2 from both sides gives:

2d^2 = 2a^2

Dividing both sides by 2 gives:

d^2 = a^2

Taking the square root of both sides gives:

d = a

So, the shortest side (a) is equal to the common difference (d). But since we have three sides in an arithmetic sequence, the shortest side will be three times the common difference.