Base Of A Right-angled Triangle Is 6 Cm More Than The Height, And The Area Is 36 Cm². (a) If The Base Is Taken As X, Express The Height In Terms Of X. (b) Find The Base And Height By Forming A Quadratic Equation.

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In the realm of geometry, right-angled triangles hold a special significance. Their unique properties and the relationships between their sides and angles make them fundamental building blocks in various mathematical and real-world applications. One common problem involving right-angled triangles is determining the lengths of their sides given certain information, such as the area and a relationship between the base and height. This article delves into such a problem, exploring how to find the base and height of a right-angled triangle when the base is 6 cm more than the height and the area is 36 cm². We will walk through the process of setting up the problem, formulating a quadratic equation, and solving it to arrive at the solution. This comprehensive guide will not only provide the answer but also enhance your understanding of the underlying mathematical principles involved.

Understanding the Problem

Before diving into the solution, it's crucial to understand the problem statement thoroughly. We are given a right-angled triangle, which means one of its angles is 90 degrees. The problem provides two key pieces of information:

  1. The base of the triangle is 6 cm more than its height.
  2. The area of the triangle is 36 cm².

Our goal is to find the lengths of the base and height of this triangle. To achieve this, we will employ algebraic techniques, specifically forming and solving a quadratic equation. By translating the given information into mathematical expressions, we can unravel the unknowns and determine the dimensions of the triangle. This problem serves as an excellent example of how algebra can be used to solve geometric problems, bridging the gap between these two branches of mathematics.

(a) Expressing Height in Terms of Base

Let's begin by expressing the height of the triangle in terms of its base. The problem states that the base is 6 cm more than the height. If we denote the base as x cm, we can represent the height as (x - 6) cm. This simple algebraic expression captures the relationship between the base and height, laying the foundation for our next steps. By defining the variables clearly and expressing the height in terms of the base, we have taken the first step towards formulating an equation that we can solve. This approach of using variables to represent unknowns is a cornerstone of algebra, allowing us to manipulate and solve for the desired quantities.

This representation is crucial because it allows us to express the area of the triangle in terms of a single variable, x. We know that the area of a triangle is given by the formula (1/2) * base * height. Substituting our expressions for base and height, we get the area as (1/2) * x * (x - 6). Since we are given the area as 36 cm², we can set up an equation that relates x to the known area. This equation will be a quadratic equation, which we can then solve to find the value of x, and consequently, the height of the triangle. The ability to translate word problems into algebraic equations is a vital skill in mathematics, enabling us to tackle a wide range of problems in various fields.

(b) Forming and Solving the Quadratic Equation

Now, let's form a second-degree equation to find the base and height. We know the area of a triangle is given by:

Area = (1/2) * base * height

Substituting the given values and our expressions for base and height, we get:

36 = (1/2) * x * (x - 6)

To solve this equation, we first multiply both sides by 2 to get rid of the fraction:

72 = x(x - 6)

Expanding the right side, we obtain a quadratic equation:

72 = x² - 6x

Rearranging the terms to set the equation to zero, we get the standard form of a quadratic equation:

x² - 6x - 72 = 0

Now, we can solve this quadratic equation using various methods, such as factoring, completing the square, or the quadratic formula. In this case, factoring is a straightforward approach. We need to find two numbers that multiply to -72 and add up to -6. These numbers are -12 and 6. Therefore, we can factor the quadratic equation as:

(x - 12)(x + 6) = 0

Setting each factor equal to zero, we get two possible solutions for x:

x - 12 = 0 => x = 12

x + 6 = 0 => x = -6

Since the base of a triangle cannot be negative, we discard the solution x = -6. Thus, the base x is 12 cm.

Now, we can find the height using the expression we derived earlier:

Height = x - 6 = 12 - 6 = 6 cm

Therefore, the base of the triangle is 12 cm, and the height is 6 cm. This solution satisfies both conditions given in the problem: the base is 6 cm more than the height, and the area is 36 cm². Solving quadratic equations is a fundamental skill in algebra, and this problem demonstrates its application in a geometric context. By carefully setting up the equation and applying factoring techniques, we were able to find the dimensions of the triangle.

Verifying the Solution

It's always a good practice to verify the solution to ensure it satisfies the original problem statement. We found the base to be 12 cm and the height to be 6 cm. Let's check if these values meet the given conditions:

  1. Is the base 6 cm more than the height? Yes, 12 cm is indeed 6 cm more than 6 cm.
  2. Is the area 36 cm²? Let's calculate the area using our values:

Area = (1/2) * base * height = (1/2) * 12 cm * 6 cm = 36 cm²

The calculated area matches the given area, so our solution is verified. This step is crucial in problem-solving as it helps to catch any errors made during the process. By plugging the solution back into the original conditions, we can gain confidence in our answer and ensure that it is correct. Verification also reinforces our understanding of the problem and the relationships between the different quantities involved.

Conclusion

In this article, we successfully solved a problem involving a right-angled triangle, finding its base and height given the area and a relationship between the sides. We started by understanding the problem, expressing the height in terms of the base, and then forming a quadratic equation. By solving the quadratic equation using factoring, we determined the base and height of the triangle. Finally, we verified our solution to ensure its accuracy.

This problem illustrates the power of algebra in solving geometric problems. By translating the given information into mathematical expressions and equations, we can unravel the unknowns and arrive at the solution. The process of setting up the equation, solving it, and verifying the solution is a fundamental approach in problem-solving that can be applied to a wide range of mathematical and real-world scenarios. Mastering these skills is essential for anyone seeking to excel in mathematics and its applications.

This exercise also highlights the importance of understanding quadratic equations and their various solution methods. Factoring, completing the square, and the quadratic formula are all valuable tools in solving quadratic equations, and choosing the most appropriate method can simplify the process. By practicing these techniques, one can develop proficiency in solving quadratic equations and apply them confidently to various problems. The ability to solve quadratic equations is a cornerstone of algebra and has applications in numerous fields, including physics, engineering, and economics.

In conclusion, solving right-angled triangle problems using quadratic equations involves a combination of geometric understanding and algebraic skills. By carefully translating the problem statement into mathematical expressions, forming an equation, solving it, and verifying the solution, we can successfully tackle such problems. This approach not only provides the answer but also enhances our problem-solving abilities and deepens our understanding of the underlying mathematical principles.