The given diagram depicts a right triangle $\triangle AB\Gamma$, with the following information provided:

• $AB = 20$ (hypotenuse),
• $A\Gamma = 15$ (one leg),
• $B\Gamma = x$ (unknown leg).

To find $x$, we can apply the Pythagorean theorem, which states: $AB^2 = A\Gamma^2 + B\Gamma^2$

Steps:

1. Substitute the known values into the equation: $20^2 = 15^2 + x^2$

2. Simplify: $400 = 225 + x^2$

3. Solve for $x^2$: $x^2 = 400 - 225 = 175$

4. Solve for $x$: $x = \sqrt{175} = \sqrt{25 \cdot 7} = 5\sqrt{7} \approx 13.23$

Final Answer:

$x = 5\sqrt{7} \approx 13.23$

Would you like a detailed explanation of the steps or any additional clarification?

Related Questions:

1. How can the Pythagorean theorem be applied in real-world situations?
2. What are the properties of right triangles in trigonometry?
3. How can you find the angles in a right triangle using trigonometric ratios?
4. What are the practical uses of surds (like $\sqrt{7}$) in geometry?
5. How can one verify the correctness of calculated triangle sides?

Tip: Always verify the largest side of a triangle as the hypotenuse to ensure correct application of the Pythagorean theorem.