Two plain mirrors incline at an unknown angle forms two images. Find the value of the angles.
Show workings step by step
To find the angle between the two mirrors, we can use the law of reflection. According to the law of reflection, the angle of incidence is equal to the angle of reflection.
Let's say the angle between the mirrors is x.
The first mirror reflects light at an angle of x/2 degrees.
The second mirror reflects light at an angle of x/2 degrees as well.
When two mirrors reflect light, the angle between the incident ray and the reflected ray is twice the angle of reflection. Therefore, the angle between the incident ray and the reflected ray from the first mirror is x degrees, and the angle between the incident ray and the reflected ray from the second mirror is also x degrees.
Now, we have two triangles formed: a triangle formed by the incident ray and the reflected ray from the first mirror, and a triangle formed by the reflected ray from the first mirror and the reflected ray from the second mirror.
Using the law of reflection, we know that the sum of the angles in a triangle is 180 degrees. Thus, we can write the following equation:
180 - (x + 90) + (x + 90) + x = 180
Simplifying this equation, we get:
180 - x - 90 + x + 90 + x = 180
Simplifying further, we have:
180 - x = 180
Subtracting 180 from both sides of the equation, we get:
-x = 0
Dividing by -1, we find:
x = 0
Therefore, the angle between the two mirrors is 0 degrees.
Please note that in this scenario, we assumed that the mirrors are positioned in a way that allows for the formation of distinct images. If the mirrors were positioned parallel to each other or at certain angles, it might not be possible to form distinct images.
To find the value of the angles formed by two inclined mirrors, we can use the concept of alternate angles and supplementary angles.
Let's assume that the angle between the mirrors is represented by theta (θ).
Step 1: Draw a diagram.
- Draw two inclined mirrors, intersecting at an angle θ.
- Label the angles and images for reference.
Step 2: Identify the angles formed by the mirrors.
- Label the top angle as angle A and the bottom angle as angle B.
- These angles are alternate angles, meaning they have the same measure.
- So, angle A = angle B = θ.
Step 3: Identify the angles formed by the reflected rays.
- Label the angle formed by the incident ray and the normal as angle X.
- The reflected ray and the normal form alternate angles with the incident ray.
- So, angle X = θ.
Step 4: Identify the angles formed by the reflected rays and the normal.
- Label the angle between the normal and the reflected ray as angle Y.
- Angle Y is supplementary to angle X.
- So, angle Y = 180° - angle X.
- Therefore, angle Y = 180° - θ.
Step 5: Identify the angles formed by the second reflected ray and the normal.
- Label the angle between the normal and the second reflected ray as angle Z.
- Angle Z is also supplementary to angle X.
- So, angle Z = 180° - angle X.
- Therefore, angle Z = 180° - θ.
Step 6: Verify the results.
- Angle X, angle Y, and angle Z should form a straight line or be equal to 180°.
- Let's check: θ + (180° - θ) + (180° - θ) = 360° - 2θ.
- Since this equals 180°, our results are verified.
Therefore, the value of the angles formed by the inclined mirrors is:
- Angle A = Angle B = θ
- Angle X = θ
- Angle Y = 180° - θ
- Angle Z = 180° - θ