Question

Find the distance between each pair of points. Round to the nearest tenth, if necessary. \(P(5,6), Q(-3,1)\)

Short answer

The distance is approximately 9.4.

Step-by-step solution

Understand the Distance Formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be found using the distance formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]. This formula is derived from the Pythagorean Theorem.

Identify the Coordinates

First, identify the coordinates of the two points given:- Point \(P\) has coordinates \((5, 6)\)- Point \(Q\) has coordinates \((-3, 1)\)

Plug Coordinates into the Formula

Substitute the coordinates of points \(P\) and \(Q\) into the distance formula. Here, \(x_1 = 5\), \(y_1 = 6\), \(x_2 = -3\), and \(y_2 = 1\):\[d = \sqrt{((-3) - 5)^2 + (1 - 6)^2}\]

Simplify the Expressions Inside the Square Root

Calculate \((-3) - 5 = -8\) and \(1 - 6 = -5\).

Square the Results

Now square each result: \((-8)^2 = 64\) and \((-5)^2 = 25\).

Add the Squared Values

Add the squared values together: \(64 + 25 = 89\).

Calculate the Square Root

Take the square root of the sum: \(\sqrt{89} \approx 9.4\).

Round to the Nearest Tenth

The final result, rounded to the nearest tenth, is \(9.4\).

Key concepts

Pythagorean theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates to the lengths of the sides of a right triangle. It states that in a right-angled triangle, the sum of the squares of the two shorter sides (often referred to as "legs") is equal to the square of the longest side (the hypotenuse). Mathematically, this can be expressed as:\[ a^2 + b^2 = c^2 \]where \( a \) and \( b \) are the lengths of the legs, and \( c \) is the length of the hypotenuse. This theorem is not only essential for solving problems involving right triangles but also acts as the foundation for the distance formula in coordinate geometry.
By using this theorem, we can determine the distance between two points by treating that distance as the hypotenuse of a right triangle formed by the horizontal and vertical distances (or "legs") between the points.

coordinate geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where algebra is used to study geometrical problems. It involves the use of coordinates to represent and analyze geometric shapes.

• Each point in the plane is represented by a pair of numbers called coordinates.
• The horizontal axis is typically labeled \( x \), and the vertical axis is labeled \( y \).
• Point \( P(5, 6) \) refers to a position 5 units along the \( x \)-axis and 6 units up on the \( y \)-axis.

This system provides an excellent way to relate algebra and geometry because it allows us to use equations to represent geometric figures and calculate various properties, such as the distance between points.
In the distance calculation between points, the coordinate geometry framework helps by identifying specific values to plug into formulas, such as the distance formula.

distance calculation

Calculating the distance between two points in a coordinate plane is a practical application of both coordinate geometry and the Pythagorean Theorem. The distance formula, derived from the Pythagorean Theorem, allows us to find this distance efficiently:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. In the example given, Points \( P(5, 6) \) and \( Q(-3, 1) \), we:

• Calculate the differences: \((-3 - 5) = -8\) and \((1 - 6) = -5\).
• Square those differences: \((-8)^2 = 64\) and \((-5)^2 = 25\).
• Add the results: \(64 + 25 = 89\).
• Finally, take the square root: \(\sqrt{89} \approx 9.4\).

These steps illustrate how the mathematical principles serve to accurately determine the distance between two points, ensuring a comprehensive understanding of the distance across a coordinate plane.