A stationary balloon is observed from three points A, B and C on the plane ground and is found that its angle of elevation from each of these points is `alpha`, if `angleABC=beta and AC=b`, the height of the balloon is____

Answer

To find the height of the balloon observed from points A, B, and C, we can follow these steps:

### Step 1: Understand the Geometry
We have three points A, B, and C on the ground, where the angles of elevation to the balloon from these points are equal to α. The angle ∠ABC is given as β, and the distance AC is given as b.

### Step 2: Set Up the Right Triangle
From point B, we can draw a vertical line to the balloon, creating a right triangle with:
- The height of the balloon (H) as the opposite side,
- The distance from point B to the point directly below the balloon (let's call it R) as the adjacent side.

### Step 3: Use the Angle of Elevation
From point B, we can use the tangent function to express the height of the balloon:
\[ \tan(\alpha) = \frac{H}{R} \]
This gives us:
\[ H = R \tan(\alpha) \quad \text{(Equation 1)} \]

### Step 4: Relate R to the Distance AC
From the triangle formed by points A, B, and C, we can apply the sine rule. The distance AC is b, and we know that:
\[ AB = R \quad \text{and} \quad BC = R \]
Thus, the total distance AC can be expressed as:
\[ AC = AB + BC = R + R = 2R \]
From this, we can express R in terms of b:
\[ R = \frac{b}{2} \quad \text{(Equation 2)} \]

### Step 5: Substitute R into the Height Equation
Now, we can substitute the value of R from Equation 2 into Equation 1:
\[ H = \left(\frac{b}{2}\right) \tan(\alpha) \]

### Step 6: Final Expression for Height
Thus, the height of the balloon can be expressed as:
\[ H = \frac{b}{2} \tan(\alpha) \]

### Summary
The height of the balloon is given by:
\[ H = \frac{b}{2} \tan(\alpha) \]