A triangle with side lengths of 5, 12, and 15 centimeters is similar to another triangle. The longest side of the other triangle has length 24 centimeters. What is the perimeter, in centimeters, of the triangle?

Options

A. `38.4`
B. `44`
C. `51.2`
D. `58`

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step 1: Identify the sides of the given triangle.
The sides of the first triangle are:
- Side 1: 5 cm
- Side 2: 12 cm
- Side 3 (longest side): 15 cm

### Step 2: Identify the longest side of the second triangle.
The longest side of the second triangle is given as 24 cm.

### Step 3: Set up the ratio of the sides of the triangles.
Since the triangles are similar, the ratio of the corresponding sides will be the same. We can set up the ratio of the longest sides:
\[ \frac{15}{24} \]

### Step 4: Simplify the ratio.
To simplify the ratio:
\[ \frac{15}{24} = \frac{5}{8} \]
This means that the sides of the second triangle are in the ratio of 5:8 compared to the first triangle.

### Step 5: Find the lengths of the other sides of the second triangle.
Using the ratio \( \frac{5}{8} \), we can find the lengths of the other sides of the second triangle.

#### For the shortest side:
Let \( x \) be the length of the shortest side in the second triangle. Using the ratio:
\[ \frac{5}{x} = \frac{5}{8} \]
Cross-multiplying gives:
\[ 5 \cdot 8 = 5 \cdot x \implies 40 = 5x \implies x = \frac{40}{5} = 8 \text{ cm} \]

#### For the middle side:
Let \( y \) be the length of the middle side in the second triangle. Using the ratio:
\[ \frac{12}{y} = \frac{5}{8} \]
Cross-multiplying gives:
\[ 12 \cdot 8 = 5 \cdot y \implies 96 = 5y \implies y = \frac{96}{5} = 19.2 \text{ cm} \]

### Step 6: List the sides of the second triangle.
Now we have all the sides of the second triangle:
- Shortest side: 8 cm
- Middle side: 19.2 cm
- Longest side: 24 cm

### Step 7: Calculate the perimeter of the second triangle.
The perimeter \( P \) of the triangle is the sum of its sides:
\[ P = 24 + 19.2 + 8 \]
Calculating this gives:
\[ P = 51.2 \text{ cm} \]

### Final Answer:
The perimeter of the triangle is **51.2 cm**.

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