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Magazine Chapter 1: Problem 19
Find the value(s) of \(k\) so that the distance between the points is \(5.\) $$(5,-1) \text { and }(k, 2)$$
Short Answer
Expert verified
The values of \( k \) are 9 and 1.
Step by step solution
01
Identify the Distance Formula
The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( ext{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). In this problem, we are given the distance as 5.
02
Substitute Known Values into the Formula
Substituting the given points into the formula, we have: \[ \sqrt{(k - 5)^2 + (2 - (-1))^2} = 5 \]. This simplifies to: \[ \sqrt{(k - 5)^2 + (2 + 1)^2} = 5 \].
03
Simplify the Equation
Calculate inside the square root: \( (2 + 1)^2 = 3^2 = 9 \), giving us the equation \[ \sqrt{(k - 5)^2 + 9} = 5 \].
04
Square Both Sides
To eliminate the square root, square both sides of the equation: \[ (k - 5)^2 + 9 = 25 \].
05
Solve for the Variable
Subtract 9 from both sides: \( (k - 5)^2 = 16 \). Taking the square root of both sides, \( k - 5 = \pm 4 \).
06
Find Possible Values of k
This gives us two equations: \( k - 5 = 4 \) and \( k - 5 = -4 \). Solve these to get the possible values of \( k \): 1. \( k = 9 \) from \( k - 5 = 4 \).2. \( k = 1 \) from \( k - 5 = -4 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solving Equations
Solving equations involves finding the value of the variable that makes a mathematical statement true. In the context of distance problems, it begins by inserting the known values into the distance formula鈥攖he initial equation given as \[\sqrt{(k - 5)^2 + (2 + 1)^2} = 5\]. By manipulating this formula, our aim is to isolate the variable, in this case, \( k \). This starts by removing the square root through squaring both sides, resulting in \[(k - 5)^2 + 9 = 25\].
- Addition or subtraction is used to eliminate numbers: subtract 9 to focus on \((k - 5)^2\).
- The goal is to simplify sides sequentially: \((k - 5)^2 = 16\).
- Finally, reverse operations to solve: square root both sides to find \(k - 5 = \pm 4\).
Distance between Points
The distance between points in a coordinate plane is calculated using a straightforward formula. This formula helps you measure the straight-line distance between two points, \((x_1, y_1)\) and \((x_2, y_2)\),and is expressed as: \[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]. In this problem, we start with the points \((5, -1)\) and \((k, 2)\).
- Replace \(x_1\) with 5 and \(y_1\) with -1.
- Replace \(x_2\) with \(k\) and \(y_2\) with 2.
Square Roots
In mathematics, a square root is a value that, when multiplied by itself, gives the original number. Calculating square roots is a critical step when dealing with distance problems. For instance, \[\sqrt{(k - 5)^2 + 9} = 5.\] This formula requires working within the root, and simplifying to make solving easier.
- First, square the known terms: \((2 + 1)^2 = 9.\)
- Then, remove the square root by squaring both sides: \((k - 5)^2 + 9 = 25.\)
Mathematical Problem Solving
Mathematical problem solving is an art and a science. It involves creativity, critical thinking, and using systematic approaches to tackle challenges. With our task of finding \( k \) when the distance is given, we use several mathematical strategies:
- Firstly, identify relevant formulas and substitute known values.
- Secondly, simplify expressions step-by-step, paying close attention to each operation's impact on the equation.
- Lastly, solve for unknowns by isolating variables and checking for multiple solutions, as in \((k - 5 = \pm 4)\) leading to \(k = 9\) or \(k = 1.\)
