In a right triangle where the sides form an arithmetic sequence, the shortest side must be less than half the length of the hypotenuse.

Here's why:

Let's denote the sides of the triangle as a, b, and c, where a

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In a right triangle where the sides form an arithmetic sequence, the shortest side must be less than half the length of the hypotenuse.

Here's why:

Let's denote the sides of the triangle as a, b, and c, where a < b < c. Since they form an arithmetic sequence, the difference between consecutive terms is constant. Therefore, we can say that b - a = c - b.

Because it's a right triangle, by the Pythagorean theorem, we know that a^2 + b^2 = c^2.

Substituting b = (a + c) / 2 into the Pythagorean theorem gives us a^2 + ((a + c) / 2)^2 = c^2. Simplifying this equation leads to a = c / 2 * sqrt(2 - sqrt(2)).

This shows that the length of the shortest side (a) is less than half the length of the hypotenuse (c).

Knowee AI · Accepted Answer

In a right triangle where the sides form an arithmetic sequence, the shortest side must be less than half the length of the hypotenuse.

Here's why:

Let's denote the sides of the triangle as a, b, and c, where a < b < c. Since they form an arithmetic sequence, the difference between consecutive terms is constant. Therefore, we can say that b - a = c - b.

Because it's a right triangle, by the Pythagorean theorem, we know that a^2 + b^2 = c^2.

Substituting b = (a + c) / 2 into the Pythagorean theorem gives us a^2 + ((a + c) / 2)^2 = c^2. Simplifying this equation leads to a = c / 2 * sqrt(2 - sqrt(2)).

This shows that the length of the shortest side (a) is less than half the length of the hypotenuse (c).

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?

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