Question

Find the distance \(d\) between the points \(P_{1}\) and \(P_{2}\). $$ P_{1}=(3,-4) ; \quad P_{2}=(5,4) $$

Short answer

The distance between the points is \( 2\sqrt{17} \).

Step-by-step solution

Identify coordinates of the points

Determine the coordinates of the given points: Point 1, \( P_{1} = (3, -4) \), and Point 2, \( P_{2} = (5, 4) \).

Recall the distance formula

The distance formula is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].

Substitute the coordinates into the formula

Substitute \( x_1 = 3 \), \( y_1 = -4 \), \( x_2 = 5 \), and \( y_2 = 4 \) into the distance formula: \[ d = \sqrt{(5 - 3)^2 + (4 - (-4))^2} \].

Simplify the expression

Perform the arithmetic inside the square root: \[ d = \sqrt{(2)^2 + (8)^2} \]. This simplifies to: \[ d = \sqrt{4 + 64} \].

Calculate the final distance

Add the values inside the square root and find the final value: \[ d = \sqrt{68} \] Which simplifies to: \[ d = 2\sqrt{17} \].

Key concepts

distance between points

To find the distance between two points, you can use a simple but powerful equation called the distance formula. This formula is derived from the Pythagorean theorem, which helps us understand the concept of distance in a coordinate plane.
First, identify the coordinates of your points. In our example, we have two points: \( P_{1} = (3, -4) \) and \( P_{2} = (5, 4) \).
The distance formula is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

• Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.
• Subtract the x-coordinates and y-coordinates as shown.

Once you substitute these values, you can simplify to find the distance. This is an essential method in both math and various applications in science and engineering.

coordinate geometry

Understanding coordinate geometry is crucial because it allows us to locate points in a two-dimensional plane using coordinates. Coordinates are written as ordered pairs \( (x, y) \), where
- \( x \) is the horizontal position (left to right), and
- \( y \) is the vertical position (up and down).
In our distance problem, we use the coordinates of Point 1 and Point 2:

• Point 1: \( P_{1} = (3, -4) \)
• Point 2: \( P_{2} = (5, 4) \)

By understanding how to plot these points on a coordinate plane and applying the distance formula, we can measure the straight line distance between them. Coordinate geometry bridges the gap between algebra and geometry, offering a precise way to describe geometric figures and their properties.

Pythagorean theorem

The Pythagorean theorem is a foundational concept in mathematics. It 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 formula is: \[ a^2 + b^2 = c^2 \]
In the context of finding distances on a coordinate plane, imagine drawing a right triangle where:

• One leg is the horizontal distance \( (x_2 - x_1) \)
• Another leg is the vertical distance \( (y_2 - y_1) \)
• The hypotenuse is the straight-line distance between the points

So when we use the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), we are essentially applying this theorem. This connection highlights the importance of understanding both algebraic and geometric perspectives to solve problems efficiently.