Problem 7
Question
Find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (-2,-6) \text { and }(3,-4) $$
Step-by-Step Solution
Verified Answer
The distance between the points (-2,-6) and (3,-4) is approximately 7.81.
1Step 1: Identify the coordinates
The two points given are (-2,-6) and (3,-4). Let \( (x_1, y_1) = (-2, -6) \) and \( (x_2, y_2) = (3, -4) \).
2Step 2: Apply the distance formula
The Euclidean distance formula is given by \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Substituting the given coordinates into this equation yields \( d = \sqrt{(3 - (-2))^2 + ((-4) - (-6))^2} \).
3Step 3: Simplify the expression
Solving inside the brackets we get \( d = \sqrt{(3 + 2)^2 + ((-4) - 2)^2} = \sqrt{5^2 + (-6)^2} \).
4Step 4: Further simplification and rounding
We can further simplify this to get \( d = \sqrt{25 + 36} = \sqrt{61} \). As the task requires rounding the answer to two decimal places, we get \( d ≈ 7.81 \).
Key Concepts
Euclidean DistanceCoordinate GeometryMath Problem Solving
Euclidean Distance
Euclidean distance is a measure of the 'straight-line' distance between two points in a Euclidean space. The concept is named after the ancient Greek mathematician Euclid. Imagine the two points as dots on a flat piece of paper, and the Euclidean distance as the length of the line you would draw directly connecting them.
To compute the Euclidean distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in a 2D space (a plane), you apply the formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Essentially, this formula represents the Pythagorean Theorem, which states that in a right-angle 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. Here, the line segment connecting the two points acts as the hypotenuse.
To compute the Euclidean distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in a 2D space (a plane), you apply the formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Essentially, this formula represents the Pythagorean Theorem, which states that in a right-angle 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. Here, the line segment connecting the two points acts as the hypotenuse.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is the study of algebraic equations to determine the positions and relationships between different geometric figures within a coordinate system. In its most common form, this system is comprised of a two-dimensional grid defined by a horizontal x-axis and a vertical y-axis.
When solving math problems involving coordinate geometry, it is crucial to understand how each point on the grid is defined by a pair of numbers, referred to as coordinates. The first number is the x-coordinate and represents the position along the x-axis, while the second number is the y-coordinate and indicates the position along the y-axis. In the case of the above exercise, the points \( (-2,-6) \) and \( (3,-4) \) represent specific locations on the 2D plane. The distance formula is one of the fundamental tools in coordinate geometry used to calculate the distance between such points.
When solving math problems involving coordinate geometry, it is crucial to understand how each point on the grid is defined by a pair of numbers, referred to as coordinates. The first number is the x-coordinate and represents the position along the x-axis, while the second number is the y-coordinate and indicates the position along the y-axis. In the case of the above exercise, the points \( (-2,-6) \) and \( (3,-4) \) represent specific locations on the 2D plane. The distance formula is one of the fundamental tools in coordinate geometry used to calculate the distance between such points.
Math Problem Solving
Math problem solving involves a series of steps that help one understand and solve mathematical questions. The process begins with understanding the problem, constructing a strategy or finding an appropriate formula (as demonstrated with the distance formula in the exercise), performing calculations correctly, and then interpreting the result.
Effective problem solving often requires recognizing patterns, applying known formulas, logical reasoning, and critical thinking. In our example, the distance formula is precisely the sort of tool that math problem solvers often use which provides a systematic approach to finding the distance between two points. Simplifying the expression and appropriately rounding off to the required level of precision concludes the problem-solving process.
Remember that practice is key to becoming proficient at solving math problems. Working through several exercises, understanding the rationale behind formulas, and familiarizing oneself with common problem types, all contribute to stronger problem-solving abilities.
Effective problem solving often requires recognizing patterns, applying known formulas, logical reasoning, and critical thinking. In our example, the distance formula is precisely the sort of tool that math problem solvers often use which provides a systematic approach to finding the distance between two points. Simplifying the expression and appropriately rounding off to the required level of precision concludes the problem-solving process.
Remember that practice is key to becoming proficient at solving math problems. Working through several exercises, understanding the rationale behind formulas, and familiarizing oneself with common problem types, all contribute to stronger problem-solving abilities.
Other exercises in this chapter
Problem 7
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