Calculate The Area Of A Right Triangle With Sides (x – 13), (x - 26), And X
This article delves into the problem of determining the area of a right triangle given its sides expressed in terms of a variable 'x'. The sides are defined as (x – 13) cm, (x – 26) cm, and x cm. We will explore the application of the Pythagorean theorem to solve for 'x', subsequently calculating the lengths of the sides and finally, the area of the triangle. This comprehensive guide provides a step-by-step solution, ensuring a clear understanding of the concepts involved.
Understanding the Problem
Before diving into the solution, let's clearly define the problem. We are given a right triangle, a triangle with one angle measuring 90 degrees. The lengths of the three sides are expressed algebraically as (x – 13), (x – 26), and x, all measured in centimeters. Our objective is to find the area of this triangle in square centimeters. To achieve this, we must first determine the value of 'x', which will then allow us to calculate the numerical lengths of the sides. Knowing the side lengths, we can then easily compute the area using the appropriate formula for a right triangle. This problem effectively combines algebraic manipulation with geometric principles, providing a good exercise in mathematical reasoning and problem-solving skills. The core of the solution lies in recognizing that in a right triangle, the Pythagorean theorem holds true, which will be our key to unlocking the value of 'x'. Remember, the Pythagorean theorem is a fundamental concept in geometry that relates the sides of a right triangle, and its accurate application is crucial for arriving at the correct answer. Also, understanding the properties of right triangles, particularly the relationship between the legs and the hypotenuse, is essential for setting up the equation correctly. The side opposite the right angle is the hypotenuse, which is the longest side, and the other two sides are the legs. The area of a right triangle can be calculated as half the product of the lengths of its two legs. By systematically applying these concepts and formulas, we can solve this problem effectively and accurately.
Applying the Pythagorean Theorem
To solve this problem, the crucial step is applying the Pythagorean Theorem. This theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs). In mathematical terms, if 'a' and 'b' are the lengths of the legs and 'c' is the length of the hypotenuse, then the theorem can be expressed as: a² + b² = c². In our case, the sides of the triangle are given as (x – 13), (x – 26), and x. Since 'x' is likely the longest side, we can assume it to be the hypotenuse. This assumption is critical because it dictates how we set up our equation. If we incorrectly identify the hypotenuse, the entire solution will be flawed. Now, we can substitute the given side lengths into the Pythagorean theorem equation. Let a = (x – 13), b = (x – 26), and c = x. Substituting these values into the equation a² + b² = c², we get: (x – 13)² + (x – 26)² = x². This equation now forms the basis for our algebraic manipulation. Our next step involves expanding the squared terms and simplifying the equation to solve for 'x'. This will require careful attention to detail to avoid errors in algebraic manipulation. The process of expanding the squares and combining like terms is fundamental to solving algebraic equations, and proficiency in these skills is essential for success in mathematics. Once we have simplified the equation, we will have a quadratic equation in terms of 'x', which we can then solve using various methods, such as factoring, completing the square, or the quadratic formula. The solutions for 'x' will give us potential values for the side lengths of the triangle, but we must remember to check these solutions to ensure they make sense in the context of the problem, as side lengths cannot be negative. Therefore, applying the Pythagorean Theorem correctly and then meticulously simplifying the resulting equation is the cornerstone of solving this problem.
Solving for x
Now, let's dive into the process of solving for x in the equation derived from the Pythagorean theorem: (x – 13)² + (x – 26)² = x². The first step involves expanding the squared terms. Recall that (a – b)² = a² – 2ab + b². Applying this to our equation, we get: (x² – 26x + 169) + (x² – 52x + 676) = x². Next, we combine like terms on the left side of the equation: 2x² – 78x + 845 = x². Now, to simplify further, we subtract x² from both sides of the equation, resulting in: x² – 78x + 845 = 0. This is a quadratic equation in the standard form ax² + bx + c = 0, where a = 1, b = -78, and c = 845. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, factoring might be a bit challenging, so let's use the quadratic formula, which states that for an equation ax² + bx + c = 0, the solutions for x are given by: x = [-b ± √(b² – 4ac)] / (2a). Substituting our values, we get: x = [78 ± √((-78)² – 4 * 1 * 845)] / (2 * 1). Simplifying the expression under the square root: (-78)² = 6084, and 4 * 1 * 845 = 3380. So, b² – 4ac = 6084 – 3380 = 2704. The square root of 2704 is 52. Therefore, x = [78 ± 52] / 2. This gives us two possible solutions for x: x = (78 + 52) / 2 = 130 / 2 = 65, and x = (78 – 52) / 2 = 26 / 2 = 13. However, we must consider the context of the problem. The side lengths are (x – 13) and (x – 26). If x = 13, then (x – 26) would be negative, which is not possible for a side length. Therefore, x = 13 is an extraneous solution. The only valid solution is x = 65. This value makes sense in the context of the problem, as it will result in positive side lengths for the triangle. The process of solving for x involved a combination of algebraic manipulation, simplification, and the application of the quadratic formula. It also highlighted the importance of checking for extraneous solutions to ensure the answer is valid within the problem's context.
Calculating the Side Lengths
With the value of x determined to be 65 cm, we can now proceed to calculate the side lengths of the right triangle. Recall that the sides are given as (x – 13) cm, (x – 26) cm, and x cm. Substituting x = 65 into these expressions, we get: Side 1: x – 13 = 65 – 13 = 52 cm. Side 2: x – 26 = 65 – 26 = 39 cm. Side 3: x = 65 cm. So, the sides of the right triangle are 52 cm, 39 cm, and 65 cm. It's a good practice to verify that these side lengths indeed form a right triangle by checking if they satisfy the Pythagorean theorem. The theorem states that a² + b² = c², where 'c' is the hypotenuse. In our case, 65 cm is the longest side, so it is likely the hypotenuse. Let's check: 52² + 39² = 2704 + 1521 = 4225. And 65² = 4225. Since 52² + 39² = 65², the side lengths do satisfy the Pythagorean theorem, confirming that we have a right triangle. Now that we have the side lengths, we can move on to calculating the area of the triangle. The area of a right triangle is given by half the product of its legs. The legs are the two shorter sides, which in this case are 52 cm and 39 cm. The hypotenuse, 65 cm, is not used in the area calculation. The calculation of the side lengths is a straightforward process of substituting the value of 'x' into the expressions for the sides. The verification step using the Pythagorean theorem is crucial to ensure that our solution is consistent with the properties of a right triangle. This step helps to catch any potential errors in the earlier steps of the solution.
Determining the Area
Finally, we arrive at the last step: determining the area of the right triangle. As established earlier, the area of a right triangle is calculated as half the product of its two legs. We have already calculated the lengths of the sides to be 52 cm, 39 cm, and 65 cm. The legs are the two shorter sides, which are 52 cm and 39 cm. Therefore, the area of the triangle can be calculated as: Area = (1/2) * base * height = (1/2) * 52 cm * 39 cm. Multiplying 52 by 39 gives us 2028. Then, taking half of 2028, we get 1014. So, the area of the right triangle is 1014 square centimeters (cm²). It is important to include the units in the final answer to provide a complete and meaningful solution. The area is measured in square units because it represents the two-dimensional space enclosed by the triangle. In summary, to find the area, we first identified the legs of the right triangle, then multiplied their lengths, and finally, divided the result by 2. This calculation provides the numerical value for the area in square centimeters. The determination of the area is a direct application of the formula for the area of a right triangle. The steps involved are simple multiplication and division, but it's essential to use the correct values for the base and height, which are the lengths of the legs. This final step concludes the solution to the problem, providing the area of the right triangle based on the given side expressions.
Conclusion
In conclusion, we have successfully determined the area of the right triangle with sides (x – 13) cm, (x – 26) cm, and x cm. We began by applying the Pythagorean theorem to establish a relationship between the sides and solve for the value of 'x', which was found to be 65 cm. Subsequently, we calculated the lengths of the sides to be 52 cm, 39 cm, and 65 cm. Finally, using the formula for the area of a right triangle, we calculated the area to be 1014 cm². This problem demonstrates the interplay between algebra and geometry, requiring a solid understanding of the Pythagorean theorem, algebraic manipulation, and the area formula for right triangles. The process involved careful attention to detail, from setting up the initial equation to checking for extraneous solutions and ensuring the final answer is expressed in the correct units. The solution highlights the importance of systematic problem-solving in mathematics, where each step builds upon the previous one to arrive at the final answer. By breaking down the problem into smaller, manageable steps, we were able to tackle it effectively and accurately. This comprehensive approach not only provides the answer but also enhances understanding of the underlying mathematical concepts and problem-solving strategies. Mastering these skills is crucial for success in mathematics and related fields. This detailed walkthrough serves as a valuable resource for students and anyone seeking to improve their problem-solving abilities in mathematics.
