We are given a right triangle $\triangle ABC$, with a circle inscribed inside it. The radius of the circle is 3 cm, and the length of side $BC$ is 8 cm. We are asked to find the perimeter of the triangle.

Step 1: Using the Inradius Formula

The formula for the area of a right triangle with an incircle (inscribed circle) is:

$A = r \times s$

where:

• $A$ is the area of the triangle,
• $r$ is the inradius (radius of the inscribed circle),
• $s$ is the semi-perimeter of the triangle.

The semi-perimeter $s$ is given by:

$s = \frac{a + b + c}{2}$

where:

• $a$, $b$, and $c$ are the lengths of the sides of the triangle.

Step 2: Using the Right Triangle Properties

Since $\triangle ABC$ is a right triangle, we can denote:

• $AB = c$ (the hypotenuse),
• $AC = b$,
• $BC = a = 8 \, \text{cm}$.

For a right triangle, the area $A$ is also:

$A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times b = 4b$

Step 3: Relation Between Inradius and Side Lengths

For a right triangle with an incircle, the inradius $r$ is given by:

$r = \frac{a + b - c}{2}$

where $a = 8$ cm (the length of side $BC$), and $r = 3$ cm (the radius of the inscribed circle). This gives us the equation:

$3 = \frac{8 + b - c}{2}$

Multiplying both sides by 2:

$6 = 8 + b - c$

$b - c = -2 \quad \text{(Equation 1)}$

Step 4: Using Pythagoras' Theorem

For a right triangle, the Pythagorean theorem holds:

$a^2 + b^2 = c^2$

Substituting $a = 8$ into this equation:

$8^2 + b^2 = c^2$

$64 + b^2 = c^2 \quad \text{(Equation 2)}$

Step 5: Solving the System of Equations

We now have two equations:

1. $b - c = -2$
2. $64 + b^2 = c^2$

From Equation 1, we can solve for $b$ in terms of $c$:

$b = c - 2$

Substitute this expression for $b$ into Equation 2:

$64 + (c - 2)^2 = c^2$

Expanding the square:

$64 + (c^2 - 4c + 4) = c^2$

Simplify:

$64 + c^2 - 4c + 4 = c^2$

Cancel out $c^2$ from both sides:

$68 - 4c = 0$

Solving for $c$:

$4c = 68$

$c = 17 \, \text{cm}$

Step 6: Finding $b$

Substitute $c = 17$ into $b = c - 2$:

$b = 17 - 2 = 15 \, \text{cm}$

Step 7: Finding the Perimeter

Now that we have $a = 8$