Question

A square and rhombus have the same base. If the rhombus is inclined at \(60^{\circ}\), find the ratio of area of square to the area of the rhombus: (a) \(2 \sqrt{3}: 3\) (b) \(1: \sqrt{3}\) (c) \(\sqrt{3}: 2\) (d) none of these

Short answer

Answer: None of these (as the correct ratio is 2:1, which is not mentioned in the given options).

Step-by-step solution

Understand the Area Formulas of a Square and Rhombus

To find the ratio of the area of a square to the area of a rhombus, we first need to know their respective area formulas. For a square with side length 's', the area formula is simply: $$Area_{square} = s^2$$ As for the rhombus, we are given the inclination angle (\(60^{\circ}\)), and as the sum of adjacent angles is \(180^{\circ}\), the other angle is \(180^{\circ}-60^{\circ}=120^{\circ}\). The area of a rhombus can be found using the following formula: $$Area_{rhombus} = d_1 \times d_2 \times \frac{1}{2}$$ where \(d_1\) and \(d_2\) are the lengths of the two diagonals of the rhombus. In our case, since the side length of the square and the base of the rhombus are the same let the side length of square and base of rhombus be 's'.

Find the Diagonals of the Rhombus

In this rhombus, one of the diagonals (let's call it \(d_1\)) is equal to the same base length s. The other diagonal will span both of the acute angles (\(60^{\circ}\)). To find the length of this diagonal \((d_2)\), we can divide the rhombus into two congruent equilateral triangles by bisecting the \(60^{\circ}\) angle. If we drop an altitude (a perpendicular line) from the bisected angle in each equilateral triangle, it will bisect the base into two segments of length \(\frac{1}{2}s\). Since these are 30-60-90 right triangles, the altitude (half of \(d_2\)) has a length of: $$\frac{1}{2}d_2 = s\sin{30^{\circ}}$$ Therefore, \(d_2 = 2s\sin{30^{\circ}}\)

Calculate the Area of the Rhombus

Now that we have \(d_1\) and \(d_2\), we can find the area of the rhombus using the formula from Step 1: $$Area_{rhombus} = s \times 2s\sin{30^{\circ}} \times \frac{1}{2}$$ Simplifying the expression, we get: $$Area_{rhombus} = s^2\sin{30^{\circ}}$$

Calculate the Ratio of the Area of the Square to the Area of the Rhombus

The ratio of the area of the square to the area of the rhombus can be found by dividing \(Area_{square}\) by \(Area_{rhombus}\): $$\frac{Area_{square}}{Area_{rhombus}} = \frac{s^2}{s^2\sin{30^{\circ}}}$$ Since the sine of \(30^{\circ}\) is \(\frac{1}{2}\), our equation simplifies to: $$\frac{Area_{square}}{Area_{rhombus}} = \frac{1}{\frac{1}{2}} = 2$$ Which means the correct answer is: (a) \(2 \sqrt{3}: 3\) As the ratio of the area of the square to the area of the rhombus is found to be 2:1, which is not present in any of the given options. Thus, the answer is: (d) none of these

Key concepts

Area of Square

The area of a square is one of the simplest geometric calculations you can encounter. All sides of a square are equal in length and form right angles. Hence, for a square, if the side length is noted as 's', you find the area by squaring this side length:

• The formula is: \(Area_{square} = s^2\)

When you multiply a number by itself, it captures the essence of a square because the length, breadth, and height measurements traditionally stick to being equal in square geometry. This makes it straightforward to remember and compute whenever you encounter problems related to squares.
The setup of these two equal sides not only solidifies the geometric symmetry but also bridges the calculations to real-world applications like determining the size of a plot of land or the space covered by a square tile.

Area of Rhombus

A rhombus extends the concept of a square into the realm of parallelograms, where opposite sides are parallel and equal in length. However, unlike a square, angles in a rhombus are not restricted to be right angles. This introduces fascinating tilts and shifts in calculations.
The formula for the area of a rhombus hinges on its diagonals. The diagonals intersect at right angles and the formula is:

• \(Area_{rhombus} = \frac{d_1 \times d_2}{2}\)

In this setup, \(d_1\) and \(d_2\) are the lengths of the diagonals. The concept of using diagonals opens another dimension in geometry by using internal measurements of a shape rather than perimeter measurements.
This formula is helpful when you think of a rhombus as two congruent triangles, which allows a breakup into simpler, more manageable parts to calculate the area effectively. This triangular separation helps you visualize how areas more complex than simple rectangles can be accurately computed.

Trigonometric Ratios

Trigonometry is the branch of mathematics dealing with the relationships between the angles and sides of triangles. It's a powerful tool applicable in geometry that extends into numerous fields, such as physics and engineering. When you have a problem involving angles, trigonometric ratios like sine, cosine, and tangent become essential.
For example, in the case of the rhombus inclined at \(60^\circ\), you utilize the sine function:

• The sine ratio: \(\sin(\theta) = \text{opposite side} / \text{hypotenuse}\)

By using trigonometric tables or functions in your calculator, you can easily find that \(\sin(30^\circ) = 0.5\). These trigonometric ratios are not only fundamental for solving triangles but also offer a deeper insight when tackling problems involving periodic phenomena, vector resolution, and more.
Knowing how to apply these ratios, you can bridge simple linear measurements to more complex angles and curvatures found in advanced geometry and applied science use cases.

Ratio of Areas

Understanding the ratio of areas involves comparing how two shapes occupy space in relation to one another. This concept is crucial in many practical applications from design to architecture, where space efficiency is key.
For a square and a rhombus sharing the same base side 's' but differing by an inclination of \(60^\circ\) in the rhombus, the understanding of area ratio taps into examining how space is re-distributed owing to the angle:

• Start by calculating each area individually using their respective formulas.
• Divide the area of one figure by the area of the other to find the ratio.

The specific solution here leads us to calculate areas and reduces by using known trigonometric values, making the computation highly insightful. This mathematical practice fosters a strategic approach in identifying options or paths in problem-solving relating to material consumption, cost-effective designs, or determining resource allocations.
The rounding off with a ratio simplifies comparison, enabling direct interpretations.