Smallest Side of Right Angled Triangle in Arithmetic Progression

Let the lengths of the sides of the right-angled triangle be $a - d$, $a$, and $a + d$, where $a > d > 0$.

Since $a + d$ is the longest side, it must be the hypotenuse of the right-angled triangle. By the Pythagorean theorem, we have: $(a - d)^2 + a^2 = (a + d)^2$

Expanding both sides of the equation: $a^2 - 2ad + d^2 + a^2 = a^2 + 2ad + d^2$

Simplifying the equation gives: $a^2 - 2ad = 2ad$ $a^2 = 4ad$

Since side length $a > 0$, we can divide both sides by $a$: $a = 4d$

Thus, the three sides of the triangle in terms of $d$ are:

• Shortest side: $a - d = 4d - d = 3d$
• Middle side: $a = 4d$
• Hypotenuse: $a + d = 4d + d = 5d$

The area of the right-angled triangle is given as $24$. The area can be expressed using the two legs of the triangle: $\text{Area} = \frac{1}{2} \times (3d) \times (4d) = 24$

$6d^2 = 24$ $d^2 = 4$

Since $d > 0$, we get: $d = 2$

Therefore, the length of the smallest side is: $\text{Smallest side} = 3d = 3 \times 2 = 6$