The ratio of the area of two isosceles triangles having the same vertical angle (i.e. angle between equal sides) is 1:4. The ratio of their heights is

Options

A. `1:4`
B. `2:5`
C. `1:2`
D. `3:4`

Answer

To find the ratio of the heights of two isosceles triangles with the same vertical angle, we can follow these steps:

### Step 1: Understand the properties of the triangles
We have two isosceles triangles, let's denote them as Triangle ABC and Triangle PQR. Both triangles have the same vertical angle, which means that the angle between the equal sides of both triangles is the same.

**Hint:** Remember that in isosceles triangles, the two sides opposite the equal angles are of the same length.

### Step 2: Set up the ratio of the areas
According to the problem, the ratio of the areas of Triangle ABC to Triangle PQR is given as 1:4. This means:
\[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle PQR} = \frac{1}{4} \]

**Hint:** The area of a triangle can be expressed in terms of its base and height.

### Step 3: Relate the areas to the heights
The area of a triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Since both triangles have the same vertical angle, we can express their areas in terms of their respective heights (H1 for Triangle ABC and H2 for Triangle PQR) and their bases (which we can denote as B1 and B2).

Thus, we can write:
\[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle PQR} = \frac{\frac{1}{2} \times B1 \times H1}{\frac{1}{2} \times B2 \times H2} \]
This simplifies to:
\[ \frac{B1 \times H1}{B2 \times H2} \]

**Hint:** The bases of the triangles can be expressed in terms of their heights due to their similarity.

### Step 4: Use the similarity of the triangles
Since Triangle ABC is similar to Triangle PQR (by the Side-Angle-Side similarity criterion), the ratio of their corresponding heights is equal to the ratio of their bases. Therefore, we can express the ratio of the areas in terms of the heights:
\[ \frac{H1}{H2} = \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle PQR} \]

### Step 5: Substitute the area ratio
From the problem, we know:
\[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle PQR} = \frac{1}{4} \]
Thus, we can write:
\[ \frac{H1}{H2} = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

**Hint:** Remember to take the square root when relating the areas to the heights.

### Step 6: Finalize the ratio of heights
Since we have \( H1 : H2 = 1 : 2 \), we can express this as:
\[ H1 : H2 = 2 : 1 \]

### Conclusion
The ratio of the heights of the two isosceles triangles is:
\[ \boxed{2 : 1} \]