Question

Find the distance \(d\) between the points \(P_{1}\) and \(P_{2}\). $$ P_{1}=(5,-2) ; \quad P_{2}=(6,1) $$

Short answer

\( \sqrt{10} \)

Step-by-step solution

- Identify coordinates

Identify the coordinates of the given points. Here, the coordinates are: \( P_{1} = (5, -2) \) \( P_{2} = (6, 1) \).

- Write the distance formula

The distance formula for two points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) is: \[d = \sqrt{(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2}\]

- Substitute coordinates into the formula

Substitute the given coordinates into the distance formula: \( x_{1} = 5, \) \( y_{1} = -2, \) \( x_{2} = 6, \) \( y_{2} = 1 \)\[ d = \sqrt{(6 - 5)^2 + (1 - (-2))^2} \]

- Calculate the differences

Calculate the differences inside the square root: \[ d = \sqrt{(1)^2 + (3)^2} \]

- Complete the squares and add

Square the differences and sum them up: \[ d = \sqrt{1 + 9} \]

- Final calculation

Find the square root of the sum: \[ d = \sqrt{10} \]

Key concepts

coordinate geometry

Coordinate geometry is a branch of mathematics that uses a coordinate system to define and represent geometric shapes and their relationships. It merges algebra and geometry to solve problems involving points, lines, and shapes in a plane. Each point in this system is defined by an ordered pair of numbers \(x, y\), known as coordinates. The x-coordinate represents the horizontal position, while the y-coordinate represents the vertical position. In coordinate geometry, you can determine the distance between points, the midpoint of a segment, and the equations of lines and curves. Understanding these principles makes it easier to solve complex geometric problems using algebraic methods.

distance between points

The distance between two points in a coordinate plane can be found using the distance formula. This formula is derived from the Pythagorean theorem and helps you calculate the straight-line distance between any two points \(P_1=(x_1, y_1)\) and \(P_2=(x_2, y_2)\). The distance formula is expressed as: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] To use the formula, follow these steps:
- Identify the coordinates of the two points.
- Plug the coordinates into the distance formula.
- Calculate the differences in the x-coordinates and y-coordinates.
- Square these differences, sum them, and then find the square root of the result.
Using this formula ensures you accurately measure the distance between any two points in a plane.

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it is expressed as: \[ a^2 + b^2 = c^2 \] Where:

• a and b are the lengths of the two legs of the triangle.
• c is the length of the hypotenuse.

This theorem is crucial in coordinate geometry because it forms the basis for the distance formula. By considering the distance between two points as the hypotenuse of a right triangle formed by their coordinates, we can apply the Pythagorean theorem to calculate the distance. The leg lengths are the differences in the x-coordinates and y-coordinates, and the hypotenuse is the distance we seek. This approach simplifies finding distances in a coordinate plane.