Question

Find the distance between the points \((-5,3)\) and \((3,1)\) : (a) \(2 \sqrt{7}\) (b) \(3 \sqrt{14}\) (c) \(5 \sqrt{17}\) (d) \(2 \sqrt{17}\)

Short answer

Answer: The distance between the points (-5,3) and (3,1) is \(2 \sqrt{17}\).

Step-by-step solution

Recall the Distance Formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: $$ D = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} $$

Identify the given points

We are given the two points \((-5,3)\) and \((3,1)\). We can consider the first point as \((x_1, y_1)\) and the second point as \((x_2, y_2)\). So, we have: $$ x_1 = -5, y_1 = 3 \\ x_2 = 3, y_2 = 1 $$

Apply the Distance Formula

Plug the values of \(x_1, y_1, x_2,\) and \(y_2\) into the distance formula: $$ D = \sqrt{(3-(-5))^2 + (1-3)^2} $$

Calculate the distance

Proceeding with the calculations, we have: $$ D = \sqrt{(3+5)^2 + (-2)^2} \\ D = \sqrt{8^2 + 4} \\ D = \sqrt{64 + 4} \\ D = \sqrt{68} $$

Simplify the result

Finally, let's simplify the distance \(D = \sqrt{68}\) by factoring it: $$ D = \sqrt{4 \cdot 17} \\ D = 2 \sqrt{17} $$ So, the distance between the points \((-5,3)\) and \((3,1)\) is \(2 \sqrt{17}\), which is option (d).

Key concepts

Coordinate Geometry

Coordinate Geometry is a fascinating branch of mathematics that combines algebra and geometry. It deals with the study of geometric figures using a coordinate system. By plotting points on a two-dimensional plane using coordinate pairs

• The horizontal line is known as the x-axis.
• The vertical line is known as the y-axis.

These axes intersect at the origin, which is the point (0,0). Every point on this plane can be described with a pair of numbers,
known as coordinates, written as \((x, y)\).
Coordinate Geometry enables us to solve geometric problems algebraically, making it easier to visualize and calculate distances, angles, and other geometric properties.
Understanding coordinate planes and the role of x and y coordinates is crucial for grasping concepts such as distance calculations.

Distance Calculation

Distance Calculation in Coordinate Geometry involves finding the length of a line segment joining two points. This is where the Distance Formula becomes vital. It's derived from the Pythagorean Theorem. Imagine a right-angled triangle formed between two points on a graph,
and you want the length of the hypotenuse. The distance between two points \((x_1, y_1)\)
and \((x_2, y_2)\) can be found by:
\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Using this formula, you can calculate how far apart the points are on the plane. Remember,

• The subtraction \(x_2 - x_1\) calculates the distance parallel to the x-axis.
• The subtraction \(y_2 - y_1\) calculates the distance parallel to the y-axis.

After squaring these lengths,
adding them together, and then taking the square root, you have the direct, shortest path between the points, also known as the Euclidean distance.

Mathematical Formulas

Mathematical Formulas are the backbone of deriving reliable calculations from known quantities. The Distance Formula is just one of many important formulas that students encounter in high school mathematics.
Formulas offer a quick way to execute complex calculations by providing a standard method. It's essential to

• Identify the variables involved.
• Understand what each variable represents.
• Understand how to manipulate these variables within the formula.

The Distance Formula specifically helps to determine the length between two coordinate points on a plane. By substituting the coordinates into the formula, students can easily find precise measurements. Formulas are a shortcut to solving problems efficiently and accurately. Knowing how to rearrange and use formulas can empower students to solve a wide range of problems beyond basic distance calculations. Understanding the role of each component in a formula is key to mastering not just coordinate geometry but all areas of mathematics.