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Magazine Chapter 5: Problem 23
A rectangle has base length \(x+3,\) altitude length \(x+1\) and diagonals of length \(2 x\) each. What are the lengths of its base, altitude, and diagonals?
Short Answer
Expert verified
Base = 8, Altitude = 6, Diagonal = 10.
Step by step solution
01
Understand the Relationship in a Rectangle
In a rectangle, the diagonal can be expressed as the hypotenuse of a right triangle, where the base and the altitude are the other two sides. The relationship can be expressed using the Pythagorean theorem: if the base is \(b\) and the altitude is \(a\), then the diagonal \(d\) is given by \(d = \sqrt{b^2 + a^2}\).
02
Express the Known Values
For this problem, the base \(b = x + 3\), the altitude \(a = x + 1\), and the diagonal \(d = 2x\). Use these expressions in the Pythagorean theorem equation: \((x + 3)^2 + (x + 1)^2 = (2x)^2\).
03
Expand the Equation
Expand each term in the equation: \((x + 3)^2 = x^2 + 6x + 9\), \((x + 1)^2 = x^2 + 2x + 1\), and \((2x)^2 = 4x^2\). Substitute these into the equation: \(x^2 + 6x + 9 + x^2 + 2x + 1 = 4x^2\).
04
Simplify the Equation
Combine like terms to simplify the equation: \(2x^2 + 8x + 10 = 4x^2\).
05
Rearrange to Form a Quadratic Equation
Rearrange the equation to bring all terms to one side: \(0 = 4x^2 - 2x^2 - 8x - 10\), which simplifies to \(0 = 2x^2 - 8x - 10\). Divide the entire equation by 2: \(0 = x^2 - 4x - 5\).
06
Solve the Quadratic Equation
Solve \(x^2 - 4x - 5 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), with \(a = 1\), \(b = -4\), and \(c = -5\). This results in \(x = \frac{4 \pm \sqrt{16 + 20}}{2}\). Simplify this to \(x = \frac{4 \pm \sqrt{36}}{2}\), leading to solutions \(x = 5\) or \(x = -1\).
07
Determine Suitable x Values
Since lengths cannot be negative, \(x = -1\) is not a valid solution. Therefore, \(x = 5\).
08
Calculate the Dimensions of the Rectangle
Substitute \(x = 5\) back into the expressions for the base, altitude, and diagonal: Base = \(5 + 3 = 8\), Altitude = \(5 + 1 = 6\), and Diagonal = \(2 \times 5 = 10\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rectangles
In geometry, a rectangle is a type of quadrilateral where all internal angles are right angles (90 degrees). The defining features are that opposite sides are equal in length, and each angle is a right angle. This makes the properties of rectangles particularly useful in various applications:
- The two pairs of opposite sides are both parallel and equal in length.
- The diagonals of a rectangle are equal in length and intersect each other in the middle.
- Due to its right angles, rectangles are a common subject for problems involving the Pythagorean theorem.
Quadratic Equations
Quadratic equations are polynomial equations of the second degree, usually in the form of \(ax^2 + bx + c = 0\). Solving these equations is a key skill in algebra, and there are several methods to do so. In the exercise, the quadratic equation formed was \(x^2 - 4x - 5 = 0\). Here's a step-by-step look at how to address these:
- Identify the coefficients: For the quadratic equation in the problem, we have \(a = 1\), \(b = -4\), and \(c = -5\).
- The most common method for solving this is the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula allows you to find the roots of any quadratic equation.
- For our equation, after applying the formula, we obtained two potential values for \(x\): 5 and -1. It's crucial to verify which of these solutions are valid in the context given.
Problem Solving
Problem solving is a critical skill that goes beyond just solving mathematical equations. It involves understanding the problem, strategizing a solution, and arriving at a logical conclusion. Here's a breakdown of the steps involved in the exercise above that you can apply in general problem solving:
- Understand the problem: First, figure out what you're being asked to find, and what information you have. For example, we knew the shapes and dimensions were related by the Pythagorean theorem.
- Develop a Strategy: Plan how to approach the problem. This could mean setting up equations based on known formulas, like how we used the Pythagorean theorem to relate different parts of the rectangle.
- Execute the plan: Solve equations using appropriate techniques. In this case, expanding, simplifying, and rearranging terms to form a quadratic equation was necessary.
- Review/Reflect: Verify the solution by considering the context. Here, we eliminated any negative solution since dimensions (lengths) cannot be negative.
