To solve this problem, let’s analyze the given information.

Given:

• Triangle $ABΓ$ is a right triangle with $\angle A = 90^\circ$.
• $AB = 16$ cm.
• $BΓ = \frac{5}{3} AΓ$.

Objective:

1. To find the lengths of sides $BΓ$ and $AΓ$.
2. To find the area of the triangle.

Solution:

Let's proceed with the calculations step-by-step.

Step 1: Set Up Variables

Let:

• $AΓ = x$.
• Then, $BΓ = \frac{5}{3} x$.

Using the Pythagorean theorem for the right triangle $ABΓ$: $AB^2 = AΓ^2 + BΓ^2$ Substitute the values: $16^2 = x^2 + \left(\frac{5}{3}x\right)^2$

Step 2: Expand and Simplify

$256 = x^2 + \frac{25}{9}x^2$ Combine terms by expressing $x^2$ as a common factor: $256 = x^2 \left(1 + \frac{25}{9}\right)$ $256 = x^2 \cdot \frac{34}{9}$

Step 3: Solve for $x^2$

Multiply both sides by $\frac{9}{34}$: $x^2 = 256 \cdot \frac{9}{34}$ $x^2 = 67.7647 \quad (\text{approx})$