The problem asks to find the side length $x$ of a right triangle where the other two sides are 12 and 35 units. This can be solved using the Pythagorean Theorem, which states:

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

Where:

• $a$ and $b$ are the legs of the right triangle
• $c$ is the hypotenuse (the longest side opposite the right angle)

From the image:

• $12$ is one leg,
• $x$ is the other leg,
• $35$ is the hypotenuse.

We apply the Pythagorean theorem:

$x^2 + 12^2 = 35^2$

First, calculate the squares: $12^2 = 144, \quad 35^2 = 1225$

Now substitute: $x^2 + 144 = 1225$

Solve for $x^2$: $x^2 = 1225 - 144 = 1081$

Finally, take the square root: $x = \sqrt{1081} \approx 32.88$

So, the side length $x$ is approximately 32.88 units.

Would you like further details on this solution?

Here are 5 questions that expand on this topic:

1. What is the Pythagorean theorem used for in geometry?
2. How do you determine which side is the hypotenuse in a right triangle?
3. Can the Pythagorean theorem be used for non-right triangles?
4. What are some real-life applications of the Pythagorean theorem?
5. How would the problem change if the value for the hypotenuse were unknown?

Tip: Always ensure you're correctly identifying the hypotenuse before applying the Pythagorean theorem.