A student notices that the shadows of a sign and a tree lie among the same line and end at the same point. The height of the sign is 2 meters and its shadow is 3 meters in lenght. The distance from the base of the sign to the base of the tree is 12 meters. How tall is the tree?

12 - 3 = 9 m.=Length of tree's shadow.

2/3 = x/9
3x = 18
X = 6 m. = Ht. of tree.

To find the height of the tree, we can consider the similar triangles formed by the tree, the sign, and their respective shadows.

Let's label the height of the tree as 'h' meters and the length of the shadow of the tree as 'x' meters.

We can set up the following proportion based on the similar triangles:

height of sign / length of sign shadow = height of tree / length of tree shadow

Substituting the given values, we have:

2 meters / 3 meters = h meters / x meters

Next, we can cross-multiply to solve for 'x':

2x = 3h

To eliminate fractions, we can divide both sides of the equation by 2:

x = (3h) / 2

Now, we can use the distance from the base of the sign to the base of the tree to find the value of 'x'. The distance is given as 12 meters, which is equal to the length of the shadow of the sign (since the shadows lie in the same line and end at the same point).

So, we have:

x = 12 meters

We can now substitute the value of 'x' back into the equation:

12 meters = (3h) / 2

To isolate 'h', we can multiply both sides of the equation by 2/3:

h = (12 meters) * (2/3)

Simplifying the expression:

h = 8 meters

Therefore, the height of the tree is 8 meters.