Askiitians Tutor Team
Last Activity: 14 Days ago
To solve this problem, we need to visualize the setup and apply some basic principles of geometry and reflection. The plane mirror is positioned parallel to a vertical wall, and we want to determine how much of the floor space between the observer and the mirror is visible to the observer after the reflection. Let's break this down step by step.
Understanding the Setup
We have a plane mirror that is 50 cm long, hung with its lower edge 50 cm above the ground. The observer's eyes are at a height of 1.8 m (or 180 cm) above the ground. The key here is to determine how far the observer can see on the floor between themselves and the mirror.
Key Measurements
- Height of the observer's eyes: 1.8 m (or 180 cm)
- Height of the lower edge of the mirror: 0.5 m (or 50 cm)
- Length of the mirror: 50 cm
Calculating the Reflection
When light reflects off a plane mirror, the angle of incidence equals the angle of reflection. This means that the line of sight from the observer's eyes to the mirror will reflect back to the ground at the same angle. To find the visible length of the floor, we need to determine how far the observer can see down to the mirror and then back to the ground.
Finding the Distance to the Mirror
First, we need to calculate the vertical distance from the observer's eyes to the top edge of the mirror. Since the mirror is 50 cm long and its lower edge is at 50 cm, the upper edge of the mirror is at:
Height of the upper edge of the mirror = 50 cm + 50 cm = 100 cm
Now, the vertical distance from the observer's eyes (180 cm) to the upper edge of the mirror (100 cm) is:
Vertical distance = 180 cm - 100 cm = 80 cm
Using Similar Triangles
Next, we can use similar triangles to find the horizontal distance from the observer to the mirror. The distance from the observer's eyes to the ground is 180 cm, and the distance from the upper edge of the mirror to the ground is 100 cm. The ratio of these distances will help us find the horizontal distance:
Let d be the horizontal distance from the observer to the mirror. The triangles formed by the observer's line of sight to the mirror and the ground are similar:
Height of observer's eyes / Horizontal distance = Height of upper edge of mirror / d
Substituting the known values:
180 / d = 100 / d
Cross-multiplying gives us:
180 * d = 100 * d
Solving for d yields:
d = (180 * d) / 100
Now, we can find the length of the floor visible to the observer. Since the mirror is 50 cm long, the visible length on the floor will be:
Visible length = 2 * d
Final Calculation
To find the exact distance, we can use the height difference and the length of the mirror. The total visible distance on the floor will be:
Visible length = 2 * (180 cm - 100 cm) = 2 * 80 cm = 160 cm
Conclusion
Therefore, the length of the floor visible to the observer after reflection from the mirror is 160 cm. This means that the observer can see a strip of the floor that is 1.6 meters long between themselves and the mirror, thanks to the principles of reflection and geometry.