The areas of two similar triangle are `81 cm^(2) and 49 cm ^(2)` respectively. If the altitude of the bigger triangle is 4.5 cm, find th corresponding altitude of the smaller triangle.

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the relationship between the areas of similar triangles
For two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides (or altitudes).

### Step 2: Set up the ratio of the areas
Let the area of the bigger triangle be \( A_1 = 81 \, \text{cm}^2 \) and the area of the smaller triangle be \( A_2 = 49 \, \text{cm}^2 \). The ratio of the areas can be expressed as:
\[ \frac{A_1}{A_2} = \frac{81}{49} \]

### Step 3: Relate the altitudes using the area ratio
Let the altitude of the smaller triangle be \( h_2 \) and the altitude of the bigger triangle be \( h_1 = 4.5 \, \text{cm} \). The ratio of the altitudes will be the square root of the ratio of the areas:
\[ \frac{h_1}{h_2} = \sqrt{\frac{A_1}{A_2}} = \sqrt{\frac{81}{49}} \]

### Step 4: Simplify the square root
Calculating the square root gives:
\[ \sqrt{\frac{81}{49}} = \frac{\sqrt{81}}{\sqrt{49}} = \frac{9}{7} \]

### Step 5: Set up the equation for the altitudes
Now we can set up the equation:
\[ \frac{4.5}{h_2} = \frac{9}{7} \]

### Step 6: Cross-multiply to solve for \( h_2 \)
Cross-multiplying gives:
\[ 4.5 \cdot 7 = 9 \cdot h_2 \]
\[ 31.5 = 9h_2 \]

### Step 7: Solve for \( h_2 \)
Now, divide both sides by 9:
\[ h_2 = \frac{31.5}{9} = 3.5 \, \text{cm} \]

### Final Answer
The corresponding altitude of the smaller triangle is \( 3.5 \, \text{cm} \).

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