Question

An acute angle made by a side of parallelogram with other pair of parallel sides is \(60^{\circ} .\) If the distance between these parallel sides is \(6 \sqrt{3}\), the other side is : (a) \(12 \mathrm{~cm}\) (b) \(12 \sqrt{3} \mathrm{~cm}\) (c) \(15 \sqrt{3} \mathrm{~cm}\) (d) none of these

Short answer

Question: In a parallelogram, one acute angle is \(60^{\circ}\) and the distance between the parallel sides is \(6\sqrt{3}\) cm. Find the length of the side adjacent to the acute angle. Answer: (a) 12 cm

Step-by-step solution

Understand the known information

We are given the acute angle (\(60^{\circ}\)) made by a side of a parallelogram with the other pair of parallel sides. The distance between these parallel sides is \(6\sqrt{3}\).

Set up a right triangle

Choose one angle of the parallelogram as the reference angle (in this case, we choose the acute angle \(60^{\circ}\) as the reference angle). Then draw an altitude line from the vertex of the reference angle to the opposite side. This will create a right triangle with a hypotenuse (the unknown side), an opposite side (same length as the distance between the parallel sides), and an adjacent side (part of the opposite side of the parallelogram).

Apply trigonometric ratios

Now we have a right triangle with a known angle (\(60^{\circ}\)) and a known opposite side (\(6\sqrt{3}\)). We can use the sine function to find the unknown hypotenuse. The sine of an angle in a right triangle is defined as the ratio between the opposite side and the hypotenuse: \(\sin(60^{\circ}) = \frac{6\sqrt{3}}{hypotenuse}\) Now, we know that the \(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\). So we have: \(\frac{\sqrt{3}}{2} = \frac{6\sqrt{3}}{hypotenuse}\)

Solve for the unknown hypotenuse

To find the unknown hypotenuse, we can rearrange the equation and solve for it: \(hypotenuse = \frac{6\sqrt{3}}{\frac{\sqrt{3}}{2}}\) Multiply both numerator and denominator by 2: \(hypotenuse = \frac{6\sqrt{3} \times 2}{\sqrt{3}}\) Simplify: \(hypotenuse = \frac{12\sqrt{3}}{\sqrt{3}}\) \(hypotenuse = 12\) So the unknown side of the parallelogram is 12 cm. Thus, the correct answer is (a) \(12 \mathrm{~cm}\).

Key concepts

Right Triangle

A right triangle is a type of triangle where one of its angles measures exactly 90 degrees. This angle is commonly referred to as the right angle. The side opposite this angle is known as the hypotenuse, which is also the longest side. The other two sides are called the opposite side and the adjacent side, depending on the angle being considered within the triangle. Right triangles have properties that make them incredibly useful in mathematics, particularly in trigonometry:

• The Pythagorean theorem applies, allowing the relation between the sides of the triangle: \[ a^2 + b^2 = c^2 \]where \(c\) is the hypotenuse.
• Trigonometric ratios (sine, cosine, tangent) are defined based on the sides of a right triangle.

Understanding right triangles is essential because many geometric shapes can be broken down into smaller right triangles, providing a pathway to solve complex problems easily.

Sine Function

The sine function is a fundamental concept in trigonometry used to understand relationships within a right triangle. The sine function of an angle in a right triangle is defined as the ratio between the length of the side opposite the angle and the length of the hypotenuse. Mathematically, it is expressed as:\[ \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \]This ratio helps in solving problems where one side of a triangle is unknown. For instance, in this exercise, the sine of 60° is used to find an unknown side of the triangle:

• Given that \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \), the equation becomes:
• \( \frac{\sqrt{3}}{2} = \frac{6\sqrt{3}}{\text{hypotenuse}} \)

Rearranging to solve for the hypotenuse reveals how practical the sine function is in simplifying and solving such problems.

Parallelogram

A parallelogram is a four-sided shape with opposite sides that are parallel and equal in length. The characteristics of a parallelogram make understanding its properties vital for solving geometric problems:

• Opposite angles in a parallelogram are equal, and consecutive angles are supplementary.
• The diagonals of a parallelogram bisect each other, but they are not necessarily equal.
• An interesting property is that you can create right triangles using the sides and diagonals as a way to calculate distances and angles inside the shape.

In the given exercise, the parallelogram is used to form a right triangle by drawing an altitude. This helps to find unknown sides using trigonometry, such as the sine function, by considering known elements such as distance and angles.

Acute Angle

An acute angle is any angle that is less than 90 degrees. These angles are common in a variety of geometrical shapes including triangles (where at least two angles are acute) and parallelograms. Knowing about acute angles helps in:

• Understanding that in any parallelogram that isn't a rectangle, angles will include acute angles.
• Implementing trigonometric functions: since the angles in trigonometric problems are often acute, it is crucial to understand how they relate to each other and the sides of a triangle.
• Forming right triangles by drawing altitudes, paying key attention to acute angles helps in the application of functions like sine, cosine, and tangent.

In the context of this exercise, recognizing the acute angle of 60° and understanding its implications in the calculations ensures that the appropriate trigonometric function (sine) is applied correctly.