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Chapter 1: Problem 18

Finding a Distance, find the distance between the points. $$ (8,5),(0,20) $$

Short Answer

Expert verified
The distance between the two points is \(\sqrt{289}\) which equals to 17 units.

Step by step solution

01

Identify the Coordinates

Identify the coordinates of the two points. Here, the coordinates for Point 1 are (8,5) and for Point 2 are (0,20). So, \(x_1 = 8\), \(y_1 = 5\), \(x_2 = 0\), and \(y_2 = 20\).
02

Substitute into the Formula

Substitute the identified coordinates into the distance formula. So in this case, we're calculating \(\sqrt{((0 - 8)^2 + (20 - 5)^2)}\)
03

Solve the Equation

Solve the equation. The distance, therefore, equals to \(\sqrt{((-8)^2 + 15^2)} = \sqrt{(64+225)} = \sqrt{289}\).

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Key Concepts

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

Coordinate Geometry
In coordinate geometry, every point on a plane is represented by an ordered pair of numbers, usually written as \( (x, y) \). This is achieved by establishing a system of horizontal and vertical lines, known as the axes (commonly the x-axis and y-axis), that intersect at a central point called the origin \( (0, 0) \). This system of axes forms a grid called the Cartesian plane.
Each point in this grid can be uniquely determined by its coordinates, which are its distance along each axis.
  • The first number, x, is the horizontal distance from the origin.
  • The second number, y, is the vertical distance from the origin.
Understanding these coordinates is crucial for solving problems in coordinate geometry, as they allow us to graphically represent equations and relationships between different points.
Distance Calculation
The distance between two points on a coordinate plane is determined by using the distance formula. This formula is an application of the Pythagorean theorem, and it enables us to measure the straight-line distance between two points.
The distance formula is expressed as: \[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] This formula calculates the hypotenuse of a right triangle formed by the points you're measuring. A step-by-step example makes it easier to understand:
  • First, identify the coordinates of each point, like \( (8, 5) \) and \( (0, 20) \).
  • Use the formula by plugging these values into it: \( \sqrt{((0 - 8)^2 + (20 - 5)^2)} \).
  • Perform calculations by solving inside the brackets, then find the square root for the final distance.
By following these steps systematically, you can easily find the distance between any two given points.
Pythagorean Theorem
The Pythagorean theorem is fundamental in geometry and is the backbone of the distance formula. It states that in a right 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. Formally, for a right triangle with sides of lengths a and b, and hypotenuse of length c, the theorem is expressed as: \[ c^2 = a^2 + b^2 \] To apply this to coordinate geometry:
  • Imagine a right triangle where the points \( x_1, y_1 \) and \( x_2, y_2 \) form a diagonal, which acts as the hypotenuse.
  • The horizontal side \( (x_2 - x_1) \) and the vertical side \( (y_2 - y_1) \) are the two legs of this right triangle.
  • By substituting these into the Pythagorean theorem, the distance formula is derived, giving us a practical method to calculate distances in a plane.
Understanding the Pythagorean theorem helps students see why the distance formula works, providing a concrete foundation on which to build more complex geometric concepts.

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