The image shows a right triangle where one of its legs is labeled as $y$, the other leg as $x$, and the hypotenuse as 13 units. Additionally, the side adjacent to the right angle and opposite to angle $y$ is 18 units.

This forms two right triangles, where the hypotenuse of the larger triangle is 18, and the smaller triangle's hypotenuse is 13.

Since this is a right triangle problem, we will apply the Pythagorean theorem to solve for $x$.

For the triangle: $c^2 = a^2 + b^2$ Where:

• $c = 13$ (the hypotenuse),
• $b = y$ (one leg),
• $a = x$ (the other leg).

We need to use additional relationships between these triangles to determine $x$ precisely.

Since the image indicates two parts, I will proceed with the calculation based on this geometric layout. Give me a moment while I perform the calculations.We can approach this problem step-by-step by analyzing the given triangle:

Step 1: Identify the right triangle

The larger triangle in the image is split into two smaller right triangles, with one side of 18 units and a common height labeled $y$, and we also see a base of $x$.

The goal is to find $x$.

Step 2: Apply Pythagoras Theorem

For the larger triangle with hypotenuse 18 and height $13$ (which forms a right angle), we use the Pythagorean theorem:

$18^2 = x^2 + 13^2$

$324 = x^2 + 169$

$x^2 = 324 - 169 = 155$

Step 3: Solve for $x$

Now, take the square root of 155 to find $x$:

$x = \sqrt{155} \approx 12.5$

Final Answer:

So, the length of $x$, rounded to the nearest tenth, is approximately 12.5 units.

Would you like further details on any steps? Feel free to ask!

Here are 5 follow-up questions to explore this further:

1. How does the Pythagorean theorem work for different types of right triangles?
2. Can the height of the triangle be used to calculate the area?
3. How would you apply trigonometry (sine, cosine, etc.) to find unknown angles in this problem?
4. What if the triangle wasn't a right triangle—how would you approach solving for unknowns?
5. How can you check if a triangle is a right triangle given only its sides?

Tip: Always ensure the triangle you're analyzing is a right triangle before applying the Pythagorean theorem.