Understanding the Lamp Post Geometry Problem

The problem describes a triangular plot ABC with given side lengths and a lamp post situated at the midpoint of side AB. We are given information about the angle the lamp post subtends at vertex B and asked to find the height of the lamp post.

Let's break down the given information:

• Triangle ABC has sides AB = 16 m, BC = 10 m, and CA = 10 m. This means triangle ABC is an isosceles triangle.
• A lamp post is at the middle point of side AB. Let's call this midpoint M.
• Since M is the midpoint of AB, the length of MB is half of AB. So, MB = \(\frac{AB}{2} = \frac{16 \text{ m}}{2} = 8 \text{ m}\).
• The lamp post subtends an angle of 45° at vertex B. Let the top of the lamp post be P. The angle subtended at B is the angle formed by the line segment from B to the base of the lamp post (BM) and the line segment from B to the top of the lamp post (BP). This is angle \(\angle PBM\). The problem states this angle is 45°.
• A lamp post is typically vertical, meaning it makes a right angle with the ground. Therefore, the triangle formed by the lamp post, the ground from M to B, and the line from P to B is a right-angled triangle, with the right angle at M (\(\angle PMB = 90^\circ\)).

Calculating the Height of the Lamp Post using Trigonometry

We have a right-angled triangle PMB, where:

• MB is the base length on the ground from the midpoint M to vertex B. We found MB = 8 m.
• PM is the height of the lamp post (which we need to find).
• \(\angle PBM = 45^\circ\) is the angle subtended by the lamp post at B.

In the right-angled triangle PMB, we can use trigonometry to relate the angle \(\angle PBM\), the opposite side PM (height of the lamp post), and the adjacent side MB.

The trigonometric ratio that relates the opposite side and the adjacent side to an angle in a right triangle is the tangent function.

\(\tan(\theta) = \frac{\text{Opposite Side}}{\text{Adjacent Side}}\)

In our triangle PMB, for angle \(\angle PBM = 45^\circ\):

• Opposite side = PM (the height of the lamp post)
• Adjacent side = MB (the distance from M to B)

So, we have:

\(\tan(45^\circ) = \frac{PM}{MB}\)

We know that the value of \(\tan(45^\circ)\) is 1.

\(1 = \frac{PM}{8 \text{ m}}\)

To find PM, we can multiply both sides of the equation by 8 m:

\(PM = 1 \times 8 \text{ m}\)

\(PM = 8 \text{ m}\)

Therefore, the height of the lamp post is 8 m.

Confirming the Lamp Post Height Calculation

The calculated height of the lamp post is 8 m. Let's check the given options.

Our calculated height matches Option 3.

Revision Table: Key Concepts for Lamp Post Problem

Additional Information: Trigonometry in Geometry Problems

Trigonometry is a powerful branch of mathematics used to study the relationships between the sides and angles of triangles. It is particularly useful in problems involving distances, heights, and angles that can be modeled using right-angled triangles.

• Sine (sin): Ratio of the opposite side to the hypotenuse.
• Cosine (cos): Ratio of the adjacent side to the hypotenuse.
• Tangent (tan): Ratio of the opposite side to the adjacent side.

These functions help us find unknown sides or angles in a right triangle if we know some other parts. For example, if you know one side and an acute angle, you can find the other sides. If you know two sides, you can find the angles.

Special angles like 0°, 30°, 45°, 60°, and 90° have standard trigonometric values that are often used in problems without requiring a calculator.

• \(\sin(30^\circ) = 1/2\), \(\cos(30^\circ) = \sqrt{3}/2\), \(\tan(30^\circ) = 1/\sqrt{3}\)
• \(\sin(45^\circ) = 1/\sqrt{2}\), \(\cos(45^\circ) = 1/\sqrt{2}\), \(\tan(45^\circ) = 1\)
• \(\sin(60^\circ) = \sqrt{3}/2\), \(\cos(60^\circ) = 1/2\), \(\tan(60^\circ) = \sqrt{3}\)

Understanding these basic trigonometric concepts is essential for solving many geometry and real-world problems involving angles and distances, such as calculating heights of buildings, distances across rivers, or angles of elevation/depression.