Find the Distance Between Two Points (-8,5) , (-9,1)
The question asks for the calculation of the distance between two specific points in a two-dimensional Cartesian coordinate system. These points are given by their x and y coordinates: the first point is at (-8,5), and the second point is at (-9,1). To find the distance, one would typically use the distance formula that is derived from the Pythagorean theorem, which calculates the straight-line (Euclidean) distance between two points in a plane.
$\left(\right. - 8 , 5 \left.\right)$,$\left(\right. - 9 , 1 \left.\right)$
Answer
Solution:
Step 1:
Apply the distance formula to calculate the distance between the two points. The formula is: $Distance = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$
Step 2:
Insert the coordinates of the given points into the formula. $Distance = \sqrt{((-9) - (-8))^{2} + (1 - 5)^{2}}$
Step 3:
Proceed with the simplification of the expression.
Step 3.1:
Calculate $-8$ multiplied by $-1$. $Distance = \sqrt{((-9) + 8)^{2} + (1 - 5)^{2}}$
Step 3.2:
Combine $-9$ and $8$. $Distance = \sqrt{(-1)^{2} + (1 - 5)^{2}}$
Step 3.3:
Square $-1$. $Distance = \sqrt{1 + (1 - 5)^{2}}$
Step 3.4:
Deduct $5$ from $1$. $Distance = \sqrt{1 + (-4)^{2}}$
Step 3.5:
Square $-4$. $Distance = \sqrt{1 + 16}$
Step 3.6:
Sum up $1$ and $16$. $Distance = \sqrt{17}$
Step 4:
Express the result in various forms. In its exact form, the distance is $\sqrt{17}$. In decimal form, it is approximately $4.12310562...$
Knowledge Notes:
The distance formula is a fundamental concept in coordinate geometry that allows us to calculate the distance between two points in a plane. The formula is derived from the Pythagorean theorem, which states that in a 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.
The distance formula is given by:
$$ \text{Distance} = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}} $$ where $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ are the coordinates of the two points.
When using the distance formula, it's important to follow the order of operations, also known as PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). This ensures that the calculation is performed correctly.
The result of the distance formula is always non-negative, as it represents the magnitude of the distance between two points, which cannot be negative. The exact form of the distance is often left in square root form to preserve precision, but it can also be approximated to a decimal value for practical purposes.
