Evan is 1.8 metres tall. He walks between two lampposts that are 5 metres apart and notices his shadow from one of the lampposts just touches the base of the other. He also notices that the ratio of his height to the height of the lamppost is 2 to 5. How far is he from each lamppost?

To solve this problem, we can use similar triangles. Let's say Evan is standing at a distance of x meters from one of the lampposts. The distance between the two lampposts is 5 meters.

Now, let's consider the triangles formed by Evan, his shadow, and the lampposts. We have two similar triangles: one formed by Evan, his shadow, and the lamppost he is standing next to, and the other formed by Evan, his shadow, and the other lamppost.

According to the problem, the ratio of Evan's height to the height of the lamppost is 2 to 5. This means that if we take the height of the lamppost he is standing next to as h, then Evan's height would be 2h.

In the first similar triangle, we have the height of the lamppost as h and the distance from Evan to the lamppost as x. In the second similar triangle, we have the height of the other lamppost as 2h and the distance from Evan to the other lamppost as (5 - x) since the total distance between the lampposts is 5 meters.

By applying the property of similar triangles, we can set up the following proportion:

h / x = 2h / (5 - x)

Now we can solve for x:

h(5 - x) = 2hx

5h - hx = 2hx

5h = 3hx

Dividing both sides by 3h:

5 = 3x

x = 5 / 3

Therefore, Evan is standing 5/3 meters away from one lamppost and (5 - x) = (5 - 5/3) meters away from the other lamppost.

Simplifying:

x = 5 / 3

x = 1.667 meters

So Evan is standing approximately 1.667 meters away from one lamppost and approximately (5 - 1.667) = 3.333 meters away from the other lamppost.