Problem 63 - A Complete Resource Book in Mathematics for JEE Main

Question

Assertion: A pole of length \(h\) stands inside a triangular plot \(A B C\) and subtends equal angles \(\alpha\) at its vertices, then \(2 h \cos \alpha \sin A=a \sin \alpha\). Reason: For circumscribed radius \(R\) of a \(\triangle A B C\), \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2 R .\)

Step-by-Step Solution

Answer

The assertion follows trigonometric identities and geometry, consistent with required circumradius relationships.

Step 1: Understand the Assertion

The assertion claims that in a triangle \(ABC\), if a pole of height \(h\) at the center subtends equal angles \(\alpha\) at the vertices, then \(2h \cos \alpha \sin A = a \sin \alpha\). This suggests a relationship between the height of the pole, the angle at each vertex, and one of the triangle's side lengths, \(a\).

Step 2: Analyze the Geometric Configuration

Since the pole subtends equal angles at each vertex, the configuration is symmetric. The 'equal angles' condition means the pole is at the in-center or a significant symmetric point. The angles at the vertices relate the height of the pole to the side of the triangle.

Step 3: Verify the Assertion using Trigonometry

For the assertion \(2h \cos \alpha \sin A = a \sin \alpha\) to hold, redisplay this relation in terms of known trigonometric identities. Note the expression is a manipulation of trigonometric properties involving height (\(h\)), angles (\(\alpha\)), and side lengths (\(a\)).

Step 4: Evaluate the Reason

The reason states a property of the circumradius \(R\) for any triangle \(ABC\): \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R\). This is the Law of Sines applied uniformly to the triangle. Check if this property supports the relationship described in the assertion.

Step 5: Relate Assertion to Circumradius

Since \(2R\) term in the reason matches the structure of the assertion's terms (with a role for \(h, \alpha\)), realize that \(R\) represents a radius related to external conditions that balance geometry around \(h\). Compare assertion's constraints with this equality to verify mathematical soundness.

Step 6: Draw the Conclusion

The trigonometric identity expresses geometric harmony arising from the triangle's configuration. Recognize consistent themes between the expression of sides, angle subtends, and radial relationships.

Key Concepts

Law of Sines

The Law of Sines is a critically important concept in trigonometry. It connects the lengths of the sides of a triangle with the sine of its angles. This law states that for any triangle with side lengths \(a\), \(b\), and \(c\) and opposite angles \(A\), \(B\), and \(C\), we have:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R\]Here, \(R\) is the circumradius of the triangle, which we'll discuss later. This equation allows us to understand how the dimensions of the triangle relate to each other through their respective angles.
Understanding this relationship is essential, especially when tackling problems involving sine values and finding unknown lengths or angles.
In our exercise, the Law of Sines helps verify the assertion about how the height of a pole relates to the triangle’s geometric properties.

Geometric Configuration

In any given geometric configuration involving a triangle, the placement of elements like a pole or a point can significantly affect the relations of angles and sides. For instance, in the current exercise, a pole stands inside the triangle and subtends equal angles at each vertex. This symmetry is beneficial because it provides a balanced configuration.

• The arrangement suggests that the pole is at an important symmetric point, like the in-center of the triangle.
• This symmetry implies that specific predictable relationships hold between geometry and trigonometry.
• These understandings help in applying the Law of Sines effectively.

Such configurations often simplify the use of mathematical concepts by revealing symmetries that wouldn't otherwise be evident.
In this context, the pole forms a central point that aids in connecting height and angle measurements across the triangle.

Circumradius

The circumradius of a triangle, denoted as \(R\), is the radius of the circle that passes through all three vertices of the triangle, known as the circumcircle. The formula \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \) indicates that the side lengths of the triangle, divided by the sines of their respective opposite angles, equal twice the circumradius.
This value is crucial in different trigonometric calculations because it provides a common measure linked with multiple triangle properties.

• The circumradius helps in verifying trigonometric relations and assertions, such as the one provided in the exercise. It connects directly to the triangle's symmetry and side lengths.
• In problems involving poles or other external points, it serves as a balancing factor, offering coherence to geometric arrangements.

Understanding circumradius helps simplify complex trigonometric problems by reducing them to manageable components, especially useful when involving symmetric geometry and relationships like the Law of Sines.