A vertical tower OP stands at the centre O of a square ABCD . Let hand b denote the length OP and AB respectively.Suppose angle APB = 60degree then the relationship between h and b can be expressed as

Question

A vertical tower OP stands at the centre O of a square ABCD . Let hand b denote the length OP and AB respectively.Suppose angle APB = 60degree then the relationship between h and b can be expressed as

Answer 1

To determine the relationship between h (height) and b (distance from the center O to the corners of the square), we can use the given angle APB.

In triangle APB, we have an angle APB = 60 degrees. This implies that triangle APB is an equilateral triangle since all angles of an equilateral triangle are equal to 60 degrees.

Now, let's draw a diagram to explain the relationship between h and b:

```
A------B
/ \
/ \
/ \
/ \
/ \
D ----------------- C
```

In the diagram, A, B, C, and D represent the corners of the square, and O represents the center of the square. The line OP is the height (h) from the center O to the top corner P of the equilateral triangle.

Since triangle APB is an equilateral triangle, all sides are equal. Let's denote the length of each side of the triangle as s.

From the center O, we can draw a line to each corner of the square (A, B, C, and D). Since the square is formed by the lines AB, BC, CD, and DA, each side of the square has a length of b.

Since triangle APB is equilateral, we can draw a perpendicular line from P to the line AB, and it will intersect at a point M.

```
A------B
/ \
M P \
/ \
/ \
D ----------------- C
```

In triangle AMP, we have a right angle at M. Therefore, we can use the Pythagorean theorem to relate the sides of the triangle:

AM^2 + MP^2 = AP^2

Since triangle AMP is a right triangle, AM is half of AB and MP is the height h.

AM = AB / 2 = b / 2

Therefore, we have:

(b / 2)^2 + h^2 = AB^2

(b / 2)^2 + h^2 = b^2

(b^2 / 4) + h^2 = b^2

h^2 = b^2 - (b^2 / 4)

h^2 = b^2 - b^2 / 4

h^2 = (3/4) * b^2

Taking the square root of both sides, we get:

h = sqrt(3/4) * b

Simplifying further, we have:

h = (sqrt(3) / 2) * b

So, the relationship between h and b, based on the given angle APB = 60 degrees, is that h is equal to (sqrt(3) / 2) times b.

Answer 2

h and b/2 are the legs of a 30-60-90 triangle

so, h = √3 * b/2