Two isosceles triangles have equal angles and their areas are in the ratio of 16 : 25. The ratio of their corresponding heights is :

Question

Accepted Answer

To determine the ratio of the corresponding heights of two isosceles triangles with equal angles and areas in the ratio of 16:25, we need to understand the relationship between the areas and the heights of similar triangles.

1. **Understanding the Relationship:**
   - When two triangles are similar (which they are, since they have equal angles), the ratio of their areas is equal to the square of the ratio of their corresponding linear dimensions (such as heights, bases, etc.).
   - If the areas of the triangles are in the ratio of 16:25, then the ratio of their corresponding linear dimensions (heights, in this case) will be the square root of the area ratio.

2. **Calculating the Ratio of Heights:**
   - The area ratio is given as 16:25.
   - To find the ratio of the heights, take the square root of each part of the ratio:
     $$
     \text{Ratio of heights} = \sqrt{16} : \sqrt{25} = 4 : 5
     $$

3. **Explanation of Why Option 1 is Correct:**
   - The calculation above shows that the ratio of the heights is 4:5, which corresponds to Option 1. This is because the square root of 16 is 4, and the square root of 25 is 5. Therefore, the correct ratio of the corresponding heights is indeed 4:5.

4. **Explanation of Why Other Options are Incorrect:**
   - **Option 2 (5:4):** This option suggests the inverse of the correct ratio. Since the area ratio is 16:25, the height ratio must be 4:5, not 5:4.
   - **Option 3 (3:4):** This option is incorrect because the square root of 16 is not 3, and the square root of 25 is not 4. The correct calculation gives a ratio of 4:5.
   - **Option 4 (4:3):** This option is incorrect because it does not reflect the correct relationship derived from the area ratio. The correct ratio, as calculated, is 4:5.

In conclusion, Option 1 (4:5) is the correct answer because it accurately represents the ratio of the corresponding heights of the two isosceles triangles based on the given area ratio of 16:25.