Problem 52 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\mathrm{ABC}\) is an equilateral triangular plot. An electric pole stands at the vertex and makes an angle of \(60^{\circ}\) at either of the other two vertices. If the height of the triangle is \(100 \mathrm{~m}\), the height of the pole is (A) \(200 \mathrm{~m}\) (B) \(100 \mathrm{~m}\) (C) \(150 \mathrm{~m}\) (D) none of these

Step-by-Step Solution

Answer

The height of the pole is none of these (173.205 m).

Step 1: Understanding the Problem

We need to find the height of a pole situated at one vertex of an equilateral triangle and inclined at an angle of \(60^\circ\) to the ground at the other two vertices. The altitude of the equilateral triangle is given as 100 m.

Step 2: Apply Pythagorean Theorem

In an equilateral triangle, the altitude divides the triangle into two 30-60-90 right triangles. For a right triangle, we can apply the Pythagorean theorem, where the hypotenuse is half of the side of the equilateral triangle, and the opposite side to 30° (altitude) is 100 m. So, let \(a\) be the side of the equilateral triangle, then \(\frac{a}{2} = \sqrt{a^2 - 100^2}\) where \(a = 200\).

Step 3: Calculate Side Length

We found that the side length \(a\) must be 200 m so we can calculate the height of the pole using the same triangle similarity.

Step 4: Calculate the Pole Height

If the pole stands at one vertex and is inclined at an angle of \(60^\circ\), the height of the pole equals twice the altitude of the triangle, because \(tan(60^\circ) = \sqrt{3}\) which is equal to the length of the opposite side over the adjacent. Therefore, the height of the pole should be \(\sqrt{3} \times 100 = 173.205\, m\). Given the answer choices, 173.205 is not listed, so it is none of these.

Key Concepts

Angle Properties

Angle properties are fundamental when understanding equilateral triangles. An equilateral triangle has three sides of equal length and three equal angles. Each angle in an equilateral triangle measures \(60^{\circ}\). This property is crucial because it determines the overall symmetry and geometry of the triangle.
In the problem, the triangle not only provides the context for a geometrical shape but also influences the angle at which the electric pole is inclined.
Since all the angles are \(60^{\circ}\), if we draw an altitude from one of the vertices to the opposite side, it divides the equilateral triangle into two \(30-60-90\) right triangles. This division is useful as it enables us to calculate various lengths within the triangle using known trigonometric relationships. This understanding of angle properties helps to simplify complex geometrical problems, making it a basic yet essential tool for solving them.

Trigonometry

Trigonometry is the study of triangles, particularly right triangles, and the relationships between their angles and sides. In the context of this exercise, we're dealing with a \(30-60-90\) right triangle, which is a byproduct of dropping an altitude from the top vertex of the equilateral triangle to the opposite side.
The \(30-60-90\) triangle is special because its side lengths are in a predictable ratio of \(1:\sqrt{3}:2\). This means the length of the hypotenuse is twice the length of the shortest side, and the length of the longer leg is \(\sqrt{3}\) times that of the shorter leg.
Understanding these ratios allows us to employ trigonometric functions, such as sine, cosine, and tangent, which are pivotal for determining unknown sides and angles. For example, in this problem, knowing that \(\tan(60^{\circ}) = \sqrt{3}\) enables us to relate the height of the pole to the given altitude of the triangle.
Trigonometry is essential in this scenario because it helps translate the equilateral triangle's properties into numerical relationships that solve real-life problems, such as calculating the height of a pole.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, stating that in a right triangle the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem can be formulated as:

• \( c^2 = a^2 + b^2 \)

In our exercise, applying the Pythagorean Theorem is necessary to find the length of the triangle's side when the altitude is given.
The altitude divides the equilateral triangle into two identical \(30-60-90\) right triangles. Using the theorem, the hypotenuse is half of what we seek—the side of the equilateral triangle—and one leg is the given altitude.

• By solving \((\frac{a}{2})^2 + 100^2 = a^2\), we can find \(a = 200\, \text{m}\).

This step is vital for drawing accurate conclusions about the electric pole's height based on geometric reasoning. By applying the calculations from the Pythagorean Theorem to trigonometry, we determine that the height of the pole is \( \sqrt{3} \times 100 \approx 173.205 \, \text{m}\). In conclusion, the Pythagorean Theorem is indispensable for unlocking key dimensions in triangular geometry and effectively solving related problems.