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

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 diag. onal (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

None of the provided options match the calculated length.

Step 1: Identify the Given Information

We have a parallelogram with the longer side being 10 cm and the shorter side being 6 cm. The angle between the longer side and the longer diagonal is \(30^{\circ}\). We need to find the length of the longer diagonal.

Step 2: Apply the Law of Cosines

In a parallelogram, the diagonal can be found using the Law of Cosines. For the longer diagonal, \( p \), opposite to the angle \(30^{\circ}\), we can write:\[ p^2 = a^2 + b^2 - 2ab \cos(\theta) \] where \(a = 10\) cm, \(b = 6\) cm, and \(\theta = 30^{\circ}\).

Step 3: Substitute the Known Values

Substitute the values into the Law of Cosines formula:\[ p^2 = 10^2 + 6^2 - 2 \times 10 \times 6 \times \cos(30^{\circ}) \]

Step 4: Calculate \(\cos(30^{\circ})\)

We know that \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\). Using this in the equation, we get:\[ p^2 = 100 + 36 - 2 \times 10 \times 6 \times \frac{\sqrt{3}}{2} \]

Step 5: Simplify the Equation

Simplify and calculate:\[ p^2 = 100 + 36 - 60 \sqrt{3} \]\[ p^2 = 136 - 60 \sqrt{3} \]

Step 6: Calculate the Length of the Diagonal

To find \(p\), take the square root of both sides:\[ p = \sqrt{136 - 60 \sqrt{3}} \]Approximating the values, match to choices provided.

Key Concepts

Parallelogram properties

A parallelogram is a four-sided figure with opposite sides that are equal and parallel. Its properties make calculations using sides and angles more predictable. Key traits include:

• Opposite sides are equal in length.
• Opposite angles are equal.
• The diagonals bisect each other.
• The sum of any two adjacent angles is \(180^\circ\).

In this exercise, knowing the lengths of the longer and shorter sides of the parallelogram helps us use specific geometric principles to find other measurements, such as the diagonal. Using these properties, you can identify specific relationships between angles and sides, critical for solving problems with unknowns. Understanding these relationships in depth helps to simplify a set of complex geometric relationships.
Summarily, the properties make them quite reliable when applying concrete formulas. Recognizing the types of angles present could greatly assist in pairing sides and angles with the correct trigonometric relations in problems.

Law of Cosines

The Law of Cosines is an extension of the Pythagorean theorem, useful in engaging non-right angled triangles by relating its sides to its angles. It is especially critical for solving problems within parallelograms which are often not right-angled figures but require consideration of angles. The formula is:

• For any triangle with sides \(a\), \(b\), and \(c\), opposite respective angles \(A\), \(B\), and \(C\), the Law is expressed as:\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]

In this exercise, post identifying parameters of the parallelogram, we employ the Law of Cosines to determine the diagonal, treating it like a side in a triangle formed within the parallelogram. Applying correct substitutions into this law allows us to calculate unknown sides effectively. It’s essential to have a strong grasp of this formula — it provides a skeletal framework for working out lengths using given angles, especially when not all angles are given or are not directly adjacent.

Trigonometry

Trigonometry focuses on the relationships between angles and lengths in triangles. In the realm of a parallelogram, which often involves angles that are not all right angles, trigonometry helps in transitioning from given angles to more usable data pieces such as side lengths or diagonal measurements.

• Its primary functions — sine, cosine, and tangent — help solve complex relationships between angles and sides.
• In this specific problem scenario, we know the angle is \(30^\circ\), and we use: \[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]

These trigonometric ratios are fundamental, especially as they repeatedly resurface when calculating using formulas such as the Law of Cosines. They’re pivotal when expressing component functions of an angle to ascertain unknown elements within triangular shapes and their combination within broader geometric structures like parallelograms.
The ability to swiftly calculate and understand trigonometric functions simplifies many a step in geometry, distinguishing whether applied actions will yield the expected results when executed, thus fitting a harmonious-solving process.