To solve for $x$, we use the property of similar triangles: corresponding sides of similar triangles are proportional.

Step 1: Set up the proportions

The sides of the triangles are given as:

• Triangle 1: $30, 24, x$
• Triangle 2: $24, 32, 40$

Match corresponding sides:

1. $\frac{30}{24} = \frac{24}{32} = \frac{x}{40}$

Step 2: Solve for $x$

Using the third proportion: $\frac{x}{40} = \frac{30}{24}$

Simplify $\frac{30}{24}$: $\frac{30}{24} = \frac{5}{4}$

So: $\frac{x}{40} = \frac{5}{4}$

Cross-multiply: $4x = 40 \cdot 5$

$4x = 200$

$x = \frac{200}{4} = 50$

Final Answer:

$x = 50$

Let me know if you need more clarification or details!

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5 Related Questions:

1. How do you verify if two triangles are similar?
2. What is the significance of proportional sides in geometry?
3. Can you solve a similar problem where angles are given instead of sides?
4. What happens if one side length of a triangle is scaled by a factor—how does it affect similarity?
5. How do you find the ratio of areas of two similar triangles?

Tip: Always check if the triangles are similar by comparing their angles or side ratios before setting up proportions.