A man looks at the reflection of the top of the lamp-post on the mirror that is 6.6 m away from the foot of the lamppost. The man’s height is 1.25 m and he is standing 2 m away from the mirror. Assuming that the mirror is placed on the ground, facing the sky and the man, and that the mirror and the lamp-post are in a same line, find the height of the lamp-post (in metres).

Correct Answer: 4.13 m

Step-by-step Explanation:

1. Understanding the Setup and Geometry:

Let us denote the lamp-post as AB, where A is the top of the lamp-post and B is its foot on the ground.

Let the man be denoted as CD, where C is his eye level (top of the man) and D is his feet standing on the ground.

Let M be the point on the ground where the mirror is placed.

According to the given information:

- Height of the man, CD = 1.25 m

- Distance of the mirror from the foot of the man, DM = 2 m

- Distance of the mirror from the foot of the lamp-post, BM = 6.6 m

- Let the height of the lamp-post be AB = h metres.

2. Applying the Laws of Reflection:

The mirror lies flat on the ground. The light ray from the top of the lamp-post (A) reflects at the mirror (M) and enters the man's eye (C).

By the law of reflection, the angle of incidence equals the angle of reflection:

$\angle AMB=\angle CMD$

Since both the lamp-post and the man stand vertically upright on the ground, the angles formed with the ground are right angles:

$\angle ABM=\angle CDM=90°$

3. Similar Triangles Property:

Comparing $\Delta ABM$ and $\Delta CDM$:

- $\angle ABM=\angle CDM=90°$

- $\angle AMB=\angle CMD$

By the Angle-Angle (AA) similarity criterion, $\Delta ABM~\Delta CDM$.

Since corresponding sides of similar triangles are proportional, we have:

$\frac{AB}{CD}=\frac{BM}{DM}$

4. Calculation:

Substitute the known values into the ratio:

$\frac{h}{1.25}=\frac{6.6}{2}$

Solve for h:

$h=1.25\times 3.3$

$h=4.125\text{ m}$

Rounding to two decimal places gives:

$h\approx 4.13\text{ m}$

Thus, the height of the lamp-post is 4.13 m.