Chapter 12: Problem 59
What is the distance between \((-6,-2)\) and \((2,4) ?\) (A) \(2 \sqrt{5}\) (B) \(2 \sqrt{7}\) (C) 10 (D) 28
Short Answer
Expert verified
The answer is (C) 10
Step by step solution
01
Identify the Coordinates
Identify the coordinates of the two points. The first point is (-6,-2) so x1=-6 and y1=-2. The second point is (2, 4) so x2=2 and y2=4.
02
Apply the Distance Formula
Now apply these coordinates to the distance formula. Substitute x2=2, y2=4, x1=-6, y1=-2 into the formula \(d = \sqrt{(x2-x1)^2 + (y2-y1)^2}\). This gives \(d = \sqrt{(2-(-6))^2 + (4-(-2))^2}\).
03
Simplify the Expression
Simplify the expression within the square root: \(d = \sqrt{(8)^2 + (6)^2} \) which further simplifies to \(d = \sqrt{64 + 36}\).
04
Calculate the Final Value
Now calculate the value under the square root. \(d = \sqrt{100}\). Take the square root to find the final value.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Coordinate Plane
Imagine a flat surface on which you can plot points, lines, and shapes. This is what mathematicians call the coordinate plane. It's a two-dimensional space defined by a horizontal axis, known as the x-axis, and a vertical axis, called the y-axis. Together, they intersect at a point termed the origin, which has the coordinates (0,0).
Each point on this plane is determined by an ordered pair of numbers, usually written as \( (x, y) \). The 'x' represents the position along the horizontal axis, and 'y' indicates the position along the vertical axis. In the context of our exercise, the points \( (-6,-2) \) and \( (2,4) \) represent specific locations on this plane. Understanding how to navigate the coordinate plane is essential for solving many problems in geometry and algebra, including calculating distances between two points.
Each point on this plane is determined by an ordered pair of numbers, usually written as \( (x, y) \). The 'x' represents the position along the horizontal axis, and 'y' indicates the position along the vertical axis. In the context of our exercise, the points \( (-6,-2) \) and \( (2,4) \) represent specific locations on this plane. Understanding how to navigate the coordinate plane is essential for solving many problems in geometry and algebra, including calculating distances between two points.
Pythagorean Theorem
The Pythagorean theorem is a fundamental principle in geometry, particularly relevant when dealing with right-angled triangles. It states that for any right-angled 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. This relationship can be captured by the formula \( a^2 + b^2 = c^2 \), where 'c' represents the length of the hypotenuse, and 'a' and 'b' represent the lengths of the other sides.
In the context of the coordinate plane, we use this theorem to calculate the distance between two points by treating the separation along x-axis and y-axis as the sides of a right-angled triangle. Hence, the distance formula we employ directly relies on the Pythagorean theorem for its derivation.
In the context of the coordinate plane, we use this theorem to calculate the distance between two points by treating the separation along x-axis and y-axis as the sides of a right-angled triangle. Hence, the distance formula we employ directly relies on the Pythagorean theorem for its derivation.
Square Root
The square root of a number is a value that, when multiplied by itself, gives the original number. It's represented by the symbol \( \sqrt{} \) and is fundamental in solving quadratic equations and finding distances in geometry. For example, if we have a number 100, the square root of 100 is 10 because \( 10 \times 10 = 100 \).
The concept of square roots directly comes into play when working with the Pythagorean theorem, as seen in our exercise. After finding the sum of the squares of the lengths of the triangle's sides, we calculate the square root of this sum to determine the distance between two points. The process of calculating square roots, while straightforward for perfect squares, may involve an approximation for other numbers or require the use of a calculator.
The concept of square roots directly comes into play when working with the Pythagorean theorem, as seen in our exercise. After finding the sum of the squares of the lengths of the triangle's sides, we calculate the square root of this sum to determine the distance between two points. The process of calculating square roots, while straightforward for perfect squares, may involve an approximation for other numbers or require the use of a calculator.
Midpoint Formula
The midpoint formula is used to find the exact center point between two coordinates on the coordinate plane. Representing the average of the x-coordinates and y-coordinates respectively, the midpoint M is found using the formula \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). This formula provides the x and y values of the midpoint, which is equally distant from both points.
While our original exercise focuses on the distance between two points rather than the midpoint, understanding the midpoint formula is valuable for various geometry problems. For instance, if we wanted to know the center of a line segment connecting our points \( (-6,-2) \) and \( (2,4) \) from the exercise, the midpoint formula would help us determine that center point.
While our original exercise focuses on the distance between two points rather than the midpoint, understanding the midpoint formula is valuable for various geometry problems. For instance, if we wanted to know the center of a line segment connecting our points \( (-6,-2) \) and \( (2,4) \) from the exercise, the midpoint formula would help us determine that center point.