In geometric optics, understanding how plane mirrors form images can often involve applying principles of geometry, particularly similar triangles. When a mirror is held a certain distance from the eye, and an object such as a tree is positioned behind the observer, the image of the tree appears within the mirror. This occurs because light rays from the top and bottom of the tree reflect off the mirror and enter the eye, creating the perception of the tree's image inside the mirror.

For example, if a mirror 5 centimeters tall is held 15 centimeters in front of the eye, and the tree is located 40 meters behind the mirror, the height of the tree can be determined using the concept of similar triangles. The key insight is that the object and its image in a plane mirror are equidistant from the mirror, meaning the tree's image appears as far behind the mirror as the tree is in front of it.

By constructing two triangles—one smaller triangle formed by the eye, the mirror, and the mirror's height, and a larger triangle formed by the eye, the tree, and the tree's height—one can use the property that the ratios of corresponding sides in similar triangles are equal. This leads to the proportion:

$$ \frac{H_{\text{tree}}}{D_{\text{tree}}} = \frac{H_{\text{mirror}}}{D_{\text{mirror}}} $$

where $H_{\text{tree}}$ is the height of the tree, $D_{\text{tree}}$ is the total distance from the eye to the tree (which is the sum of the distance from the eye to the mirror and the distance from the mirror to the tree), $H_{\text{mirror}}$ is the height of the mirror, and $D_{\text{mirror}}$ is the distance from the eye to the mirror.

Converting all measurements to consistent units (meters), the height of the tree can be calculated by rearranging the formula:

$$ H_{\text{tree}} = \frac{H_{\text{mirror}}}{D_{\text{mirror}}} \times D_{\text{tree}} $$

Substituting the values $H_{\text{mirror}} = 0.05\, \text{m}$, $D_{\text{mirror}} = 0.15\, \text{m}$, and $D_{\text{tree}} = 40 + 0.15 = 40.15\, \text{m}$, the height of the tree is approximately 13.4 meters. This result aligns with expectations for a tall tree viewed from a significant distance through a small mirror.

This approach highlights the importance of combining geometric optics principles with fundamental geometry to solve real-world problems involving plane mirrors. Recognizing similar triangles and setting up proportional relationships allows for the determination of unknown distances or heights without direct measurement, reinforcing the interconnectedness of optics and geometry.

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