A person sits in a chair 3m from a wall on which hangs a plane mirror.Three meters directly behind him stands a woman 1.6m tall.What minimum height of plane mirror on the wall will enable him to see the woman's full height?

To determine the minimum height of the plane mirror that will enable the person to see the woman's full height, we can use the concept of similar triangles and the law of reflection.

Let's represent the distances and heights involved in the problem:

Distance between the person and the mirror: 3m
Distance between the mirror and the woman: 3m
Height of the woman: 1.6m
Height of the mirror: h (to be determined)

Now, let's consider the triangle formed by the person, the mirror, and the woman:

```
_______________
| |
| P |
| * |
| |\ |
|______|_\______|
|______|_______|
| M |
|______________|
|______________|
W
```

Applying the law of reflection, we know that the angle of incidence (θi) is equal to the angle of reflection (θr). Since the person is looking at the mirror, the angle of incidence can be formed between the person's line of sight and the mirror's surface.

Moreover, we can see two similar triangles formed between the person, the mirror, and the woman. It involves the height of the person, the distance from the person to the mirror, the height of the woman, and the distance from the mirror to the woman.

Using the properties of similar triangles, we can set up the following proportion:

(person's height) / (distance to mirror) = (woman's height) / (distance to woman)

Substituting the known values, we have:

(person's height) / 3 = 1.6 / 3

Simplifying this equation, we find that the person's height is irrelevant to the final answer.

1 = 1.6 / 3

Now, we can solve for the woman's height required for the person to see her fully. Rearranging the equation, we get:

woman's height = (distance to woman) * (woman's height) / (distance to mirror)

woman's height = 3 * 1.6 / 3

woman's height = 1.6

Therefore, the minimum height of the plane mirror on the wall should be at least 1.6 meters for the person to see the woman's full height.