Find the missing side angle or value of the variable x , for the smaller triangle a=5 b=4 and c=3 for the bigger triangle d=15 and e=12 find x

We have two triangles: a smaller triangle with sides \( a = 5 \), \( b = 4 \), and \( c = 3 \), and a larger triangle with two given sides \( d = 15 \) and \( e = 12 \), while we are tasked to find the value of \( x \), which represents the third side of the larger triangle.

Since both triangles share the same shape (they are similar), we can apply the concept of similarity, which implies that corresponding sides of similar triangles are proportional.

### Step 1: Set up the proportion
Since the triangles are similar, we can write the proportion as:
\[
\frac{a}{d} = \frac{b}{e} = \frac{c}{x}
\]
Substitute the known values:
\[
\frac{5}{15} = \frac{4}{12} = \frac{3}{x}
\]

### Step 2: Simplify the ratios
Simplifying both sides:
\[
\frac{5}{15} = \frac{1}{3}, \quad \frac{4}{12} = \frac{1}{3}
\]
Thus, we have:
\[
\frac{1}{3} = \frac{3}{x}
\]

### Step 3: Solve for \( x \)
Now solve for \( x \):
\[
\frac{1}{3} = \frac{3}{x}
\]
Cross-multiply:
\[
x = 9
\]

### Final Answer:
The missing side \( x \) for the larger triangle is \( 9 \).

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Would you like any further explanation, or do you have any questions about this problem?

Here are some related questions to expand on this topic:
1. How do you determine if two triangles are similar?
2. What are the properties of similar triangles?
3. Can you use the Pythagorean theorem to check the validity of the triangle's sides?
4. How would you solve this if the triangles weren't similar?
5. What happens to the angles when two triangles are similar?

**Tip:** When dealing with similar figures, always check for corresponding sides and angles to set up accurate proportions.

Find the missing side angle or value of the variable x, for the smaller triangle a=5, b=4, and c=3. For the bigger triangle d=15 and e=12, find x.

Learn how to find the missing side of a larger triangle when two triangles are similar, using proportionality of corresponding sides. This geometry problem involves solving for x when a=5, b=4, c=3 in the smaller triangle, and d=15, e=12 in the larger triangle. Suitable for Grades 9-10.

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