Question

In a rectangle the ratio of the length is to breadth is same as that of the sum of the length and breadth to the length. [f 1 a.d \(b\) be the length and breadth of the rectangle then whith of the following is true? (i) \(\frac{l}{b}=\frac{l^{2}}{b^{2}}+1\) (ii) \(\frac{b}{l-b}=\frac{l+b}{l}\) (iii) \(b=(l+b)(l-b)\) (a) only (i) is true (b) only (ii) is true (c) only (ii) and (iii) are true (d) only (i) and (ii) are true

Short answer

(a) (i) \(\frac{l}{b}=\frac{l^{2}}{b^{2}}+1\) (b) (ii) \(\frac{b}{l-b}=\frac{l+b}{l}\) (c) (iii) \(b=(l+b)(l-b)\) Answer: (b) Only (ii) is true

Step-by-step solution

Rewrite the given ratio

Rewrite the given ratio, multiplying both sides by \(l * b\) to eliminate the denominators: \(l^2 = b (l + b)\) Now, we will test each of the given statements.

Test statement (i)

According to statement (i): \(\frac{l}{b}=\frac{l^{2}}{b^{2}}+1\) First, cross-multiply the fractions: \(l^3 = l^2 * b^2 + l * b^2\) Now, substitute the equation \(l^2 = b (l + b)\) : \(l^3 = b (l + b)(b) + l * b^2\) Since this equation cannot be simplified further and does not hold true for all rectangles, statement (i) is false.

Test statement (ii)

According to statement (ii): \(\frac{b}{l-b}=\frac{l+b}{l}\) Take the equation derived from the given ratio, and solve for \(l - b\). After dividing both sides by 'b', we get: \(l - b = \frac{l^2}{l + b}\) Cross-multiply the fractions: \(b(l - b) = l^2\) This equation is true as it matches the original equation derived from the given ratio. Therefore, statement (ii) is true.

Test statement (iii)

According to statement (iii): \(b=(l+b)(l-b)\) Substitute the equation \(l^2 = b (l + b)\) \(l^2 = (l+b)(l-b)\) The left-hand side of this statement matches the equation derived from the given ratio. However, the right-hand side does not match. Therefore, statement (iii) is false. Based on our analysis, the correct answer is: (b) Only (ii) is true

Key concepts

Rectangles

When we think of rectangles, we imagine a four-sided shape characterized by right angles and opposite sides that are parallel and of equal length. In simple terms, a rectangle has two dimensions: length and breadth. Understanding the properties of rectangles is important for solving many geometric problems.
The length is often the longer side of the rectangle, while the breadth (or width) is the shorter one. These two elements are crucial when determining aspects such as the area and perimeter:

• The area of a rectangle is given by multiplying the length by the breadth: \( \text{Area} = l \times b \, \).
• The perimeter is the sum of all sides: \( \text{Perimeter} = 2l + 2b \, \).

When dealing with rectangle-related problems, knowing how to work with equations and ratios can help find unknown dimensions when some variables are missing. This involves the use of basic algebraic techniques, which are crucial for solving these problems in an accurate and efficient way.

Algebraic Equations

Algebraic equations are mathematical statements that show the relationship between different variables. In a rectangle problem, we use algebraic equations to represent relationships between the length and breadth, especially when they are given in a certain ratio.
For instance, if the length of a rectangle is twice its breadth, this can be expressed as:

• \( l = 2b \)

In the exercise provided, we encountered an algebraic equation stemming from a given ratio:

• \( l^2 = b(l+b) \)

This equation derived from the ratio of the rectangle's dimensions helps us verify various statements about the relationships between the length and breadth. Solving such algebraic equations typically involves rearranging terms, factoring, and using cross-multiplication to either prove or disprove the given equations.
These techniques are what allow us to manipulate and simplify equations, providing clarity and setting the foundation for uncovering the truth behind given statements.

Mathematical Reasoning

Mathematical reasoning is the process of using logical thinking and deduction to arrive at solutions. When analyzing problems involving ratios and proportions, especially in geometry, this becomes vital in justifying results.
For example, in the provided exercise: - We're given a ratio that links the dimensions of a rectangle. By manipulating the terms, we develop an equation in step 1.
When this equation is applied across different test statements, we use logical reasoning to determine which statement holds true. This involves:

• Substitute values to simplify equations.
• Analyze whether the simplified equation reflects reality.
• Determine consistency with the original equation.

In step 2 and on, reasoning allows us to test and validate the truthfulness of each statement regarding the rectangle's proportion. This helps ensure we only conclude based on correct mathematical evidence. Mathematical reasoning is essential not only for verifying statements like those in the exercise but also for solving complex problems across various domains.