The image shows a triangle with two segments divided by a line, and the problem is to calculate the missing length, denoted as $x$, with measurements in meters.

Given Information:

• The larger triangle has a height of 24 meters and a hypotenuse of 30 meters.
• The smaller triangle (inside the larger one) has a height of 18 meters.
• The two triangles are similar (indicated by the markings on the sides), meaning their sides are proportional.

Approach:

Since the triangles are similar, the ratio of corresponding sides will be equal. We can set up a proportion between the smaller and larger triangles.

For the height: $\frac{18}{24}$ For the hypotenuse (we are trying to find $x$): $\frac{x}{30}$

Now, solve for $x$:

$\frac{18}{24} = \frac{x}{30}$

Simplify $\frac{18}{24}$ to $\frac{3}{4}$, and then cross-multiply:

$\frac{3}{4} = \frac{x}{30}$

$3 \times 30 = 4 \times x$

$90 = 4x$

Solve for $x$:

$x = \frac{90}{4} = 22.5$

Conclusion:

The missing length $x$ is 22.5 meters.

Let me know if you want further details or explanations!

Related Questions:

1. How do you prove that two triangles are similar?
2. What are the properties of similar triangles?
3. Can we use another method like the Pythagorean theorem in this scenario?
4. How do you apply proportions to real-world measurement problems?
5. What are other examples where similar triangles are used in geometry?

Tip:

When working with similar triangles, always check if the corresponding angles are equal, as this confirms similarity. Then use side ratios to solve for unknowns.