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Chapter 4: Problem 3

Find two choices for \(t\) such that the distance between (2,-1) and \((t, 3)\) equals 7 .

Short Answer

Expert verified
The two values for \(t\) are 7 and -3.

Step by step solution

01

Write down the distance formula

The distance formula for two points (x1, y1) and (x2, y2) is \[D = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}\] In our case, (x1, y1) = (2, -1) and (x2, y2) = (t, 3) and we are given the distance D = 7.
02

Plug in the values and solve for t

Plugging in the values into the distance formula and solving for t, we have: \[7 = \sqrt{(t - 2)^2 + (3 - (-1))^2}\] Now, simplify and solve for t: \[t = 2 \pm 5\]
03

Find the two values of t

We have two solutions for t: 1. Add the value: \(t = 2 + 5 = 7\) 2. Subtract the value: \(t = 2 - 5 = -3\) Therefore, the two values for t are 7 and -3, which will result in a distance of 7 between the points (2, -1) and (t, 3).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Precalculus
Precalculus covers a range of topics that prepare students for the study of calculus. It includes foundations in algebra, geometry, and functions that are critical for understanding higher-level concepts. For example, when dealing with distances in a coordinate system, a strong precalculus background helps in understanding the geometric interpretation of problems as well as the algebraic manipulation involved.
Solving for variable
Solving for a variable is a fundamental skill in algebra and precalculus. It becomes particularly interesting when there are two values that satisfy an equation, which is often the case in quadratic equations or problems involving distances, like the one in our exercise. Here, students must isolate the variable, consider the square roots that could be positive or negative, and understand that each solution must be tested in the context of the problem to ensure it makes sense.
Euclidean distance
The concept of Euclidean distance stems from the Euclidean geometry, where the distance between two points is the length of the shortest path connecting them, which is the straight line. This distance can be computed in a coordinate plane using the distance formula, by taking the square root of the sum of the squares of differences in their respective x and y coordinates. This ensures that the distance is always a positive value or zero, reflecting the non-negative nature of physical distances.
Square root
The square root function is critical in many areas of mathematics, including geometry and algebra. It is the inverse operation of squaring a number. When dealing with distances or areas, the square root is often used to return to a one-dimensional measure from a squared quantity. It's important for students to remember that every positive number actually has two square roots: a positive and negative root. However, when dealing with distances, only the positive root represents a true distance value.

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