## To find the height of the tower, we can set up two equations using the tangent function:

From the first triangle, we have tan(56) = x / (20 + y), where x is the height of the tower and y is the distance between the foot of the tower and the first point.

From the second triangle, we have tan(63) = x / y.

We now have two equations and two unknowns, x and y. We can solve these equations simultaneously to find the values of x and y.

First, rearrange the equations to isolate x:

In the first equation, multiply both sides by (20 + y):

tan(56) = x / (20 + y)

x = (20 + y) * tan(56)

In the second equation, multiply both sides by y:

tan(63) = x / y

x = y * tan(63)

Now that we have two expressions for x, we can set them equal to each other:

(20 + y) * tan(56) = y * tan(63)

Next, distribute and combine like terms:

20 * tan(56) + y * tan(56) = y * tan(63)

Now, isolate y by subtracting y * tan(63) from both sides:

20 * tan(56) = y * tan(63) - y * tan(56)

Factor out y on the right side:

20 * tan(56) = y * (tan(63) - tan(56))

Finally, divide both sides by (tan(63) - tan(56)):

y = (20 * tan(56)) / (tan(63) - tan(56))

Now substitute this value of y into either of the original equations to solve for x:

x = y * tan(63)

x = [(20 * tan(56)) / (tan(63) - tan(56))] * tan(63)

Simplify and evaluate the expression to find the value of x, which represents the height of the tower.