Areas of two similar triangles are 98 sq. cm and 128 sq. cm. Find the ratio between the lengths of their corresponding sides. 

Answer

To find the ratio between the lengths of the corresponding sides of two similar triangles with given areas, follow these steps:

### Step 1: Understand the relationship between the areas and the sides of similar triangles.
The areas of two similar triangles are proportional to the squares of the lengths of their corresponding sides. This means:
\[ \frac{\text{Area}_1}{\text{Area}_2} = \left(\frac{\text{Side}_1}{\text{Side}_2}\right)^2 \]

### Step 2: Write down the areas of the triangles.
Given:
- Area of Triangle 1 = 98 sq. cm
- Area of Triangle 2 = 128 sq. cm

### Step 3: Set up the ratio of the areas.
Using the areas, we can set up the ratio:
\[ \frac{98}{128} \]

### Step 4: Simplify the ratio of the areas.
To simplify \(\frac{98}{128}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 2:
\[ \frac{98 \div 2}{128 \div 2} = \frac{49}{64} \]

### Step 5: Find the ratio of the sides.
Since the ratio of the areas is equal to the square of the ratio of the sides, we have:
\[ \frac{49}{64} = \left(\frac{\text{Side}_1}{\text{Side}_2}\right)^2 \]

### Step 6: Take the square root to find the ratio of the sides.
To find the ratio of the sides, take the square root of both sides:
\[ \frac{\text{Side}_1}{\text{Side}_2} = \sqrt{\frac{49}{64}} = \frac{\sqrt{49}}{\sqrt{64}} = \frac{7}{8} \]

### Final Answer:
The ratio between the lengths of their corresponding sides is:
\[ \frac{7}{8} \]

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