The ratio between the areas of two similar triangles is 16 : 25. Find the ratio between their :
corresponding medians. 

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between areas and sides of similar triangles.
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

### Step 2: Set up the ratio of the areas.
We are given that the ratio of the areas of the two triangles is 16:25.

### Step 3: Relate the area ratio to the side ratio.
Let the ratio of the corresponding sides of the triangles be \( a:b \). According to the relationship mentioned, we have:
\[ \frac{Area_1}{Area_2} = \left(\frac{a}{b}\right)^2 \]
Given that:
\[ \frac{Area_1}{Area_2} = \frac{16}{25} \]
We can write:
\[ \left(\frac{a}{b}\right)^2 = \frac{16}{25} \]

### Step 4: Take the square root to find the ratio of the sides.
Taking the square root of both sides gives us:
\[ \frac{a}{b} = \sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5} \]

### Step 5: Determine the ratio of the corresponding medians.
The ratio of the corresponding medians of similar triangles is the same as the ratio of their corresponding sides. Therefore, the ratio of the medians is also:
\[ \frac{4}{5} \]

### Final Answer:
The ratio between the corresponding medians of the two triangles is \( 4:5 \).

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