The areas of the similar triangles are the ratio of `25:36`. What is the ratio of their respectively heights :

Options

A. `5:6`
B. `6:5`
C. `1:11`
D. `2:3`

Answer

To find the ratio of the heights of two similar triangles given the ratio of their areas, we can follow these steps:

### Step 1: Understand the relationship between area and height in similar triangles.
The areas of similar triangles are proportional to the square of the ratio of their corresponding sides (or heights). If the ratio of the areas of two triangles is \( A_1 : A_2 \), then the ratio of their corresponding heights \( h_1 : h_2 \) can be found using the formula:
\[ \frac{h_1}{h_2} = \sqrt{\frac{A_1}{A_2}} \]

### Step 2: Identify the given ratio of areas.
From the problem, we know that the ratio of the areas of the two triangles is:
\[ A_1 : A_2 = 25 : 36 \]

### Step 3: Apply the formula to find the ratio of heights.
Using the formula from Step 1, we can substitute the area ratio:
\[ \frac{h_1}{h_2} = \sqrt{\frac{25}{36}} \]

### Step 4: Simplify the square root.
Calculating the square root gives:
\[ \frac{h_1}{h_2} = \frac{\sqrt{25}}{\sqrt{36}} = \frac{5}{6} \]

### Step 5: State the final ratio of heights.
Thus, the ratio of the heights of the two triangles is:
\[ h_1 : h_2 = 5 : 6 \]

### Summary
The ratio of the respective heights of the two similar triangles is \( 5 : 6 \).

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