Problem 34 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The longer side of a parallelogram is \(10 \mathrm{~cm}\) and the shorter is \(6 \mathrm{~cm}\). If the longer diagonal makes an angle \(30^{\circ}\) with the longer side, the length of the longer diagonal (in \(\mathrm{cm}\) ) is (A) \(5 \sqrt{3}+\sqrt{11}\) (B) \(4 \sqrt{3}+\sqrt{11}\) (C) \(5 \sqrt{3}+\sqrt{13}\) (D) none of these

Step-by-Step Solution

Answer

The correct option is not present in the list provided; none of these fit the expression derived.

Step 1: Identify Known Values

We know the longer side of the parallelogram is 10 cm, the shorter side is 6 cm, and the angle between the longer side and the diagonal is 30 degrees.

Step 2: Use the Law of Cosines

To find the length of the diagonal, we will use the Law of Cosines: \[d^2 = a^2 + b^2 - 2ab\cos(C)\]where \(d\) is the diagonal, \(a\) and \(b\) are the sides, and \(C\) is the angle between them.

Step 3: Plug in Values

Substitute the known values into the formula: \[d^2 = 10^2 + 6^2 - 2 \times 10 \times 6 \times \cos(30^{\circ})\]Simplify the terms: \[d^2 = 100 + 36 - 120 \times \frac{\sqrt{3}}{2}\]

Step 4: Simplify the Equation

Continue simplifying: \[d^2 = 136 - 60\sqrt{3}\]

Step 5: Calculate the Length of Diagonal

Calculate the roots of the equation:\[d = \sqrt{136 - 60\sqrt{3}}\]Simplify to get the length of the diagonal.

Step 6: Determine the Answer

Upon further simplification and estimation, the expression does not neatly fit any of the given options directly without additional work. So you check for possible alternatives based on given options.

Key Concepts

Law of Cosines

The Law of Cosines is an essential tool in geometry, particularly useful when you have to deal with non-right triangles. It allows us to find the length of a side when we know the lengths of two sides and the angle between them. The formula is expressed as:\[d^2 = a^2 + b^2 - 2ab\cos(C)\]Here:

• \(d\) is the side we want to determine.
• \(a\) and \(b\) are the known sides.
• \(C\) is the angle opposite side \(d\).

This formula generalizes the Pythagorean theorem, applying to any triangle, not just right-angled ones.

In the problem, we know the sides are \(10 \, \mathrm{cm}\) and \(6 \, \mathrm{cm}\), with a \(30^{\circ}\) angle. By plugging these values into the formula, we can efficiently solve for the unknown diagonal. Calculating follows the simplification process, allowing us to understand how angles affect side lengths.

Parallelogram

A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length. In this context, understanding the properties of parallelograms is crucial to solving problems about their angles and sides.

Key characteristics of a parallelogram include:

• Opposite sides are equal and parallel.
• Opposite angles are equal.
• The diagonals bisect each other.

In this exercise, we focus on the relationship between the sides and diagonals. The diagonal creates two triangles within the parallelogram, each sharing a side with the original shape. This division allows us to apply trigonometric principles such as the Law of Cosines to determine unknown measurements.

The angles and diagonals are particularly useful when solving for lengths as they add additional reference points, helping to understand the spatial arrangement and symmetry inherent in parallelograms.

Trigonometry

Trigonometry is a branch of mathematics that examines the relationships between side lengths and angles in triangles. It's particularly useful for solving problems involving distances and angles. Core trigonometric functions include sine, cosine, and tangent.

In our exercise involving the parallelogram, the cosine function is used to assess the angle relationship, specifically the \(30^{\circ}\) angle in the diagram. Understanding the cosine function's role helps in calculating the length of the diagonal accurately:

• The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse.
• For a \(30^{\circ}\) angle, \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\).

By incorporating this trigonometric value into the Law of Cosines formula, we determine the length of the diagonal more effectively. Trigonometry provides the mathematical tools necessary to bridge angles and side lengths in geometric figures beyond simple calculations, enhancing problem-solving skills in complex shapes like parallelograms.