Problem 3 - Algebra and Trigonometry

Question

The distance between the points \((a, b)\) and \((c, d)\) is ________. So the distance between \((1,2)\) and \((7,10)\) is ________.

Step-by-Step Solution

Answer

The distance is 10 units.

Step 1: Understanding the distance formula

To calculate the distance between two points in a coordinate plane, use the distance formula: \( D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Here, \((x_1, y_1)\) are the coordinates of the first point and \((x_2, y_2)\) are for the second point.

Step 2: Substituting the given values

The given points are \((1, 2)\) and \((7, 10)\). Substitute these into the formula: \( x_1 = 1, y_1 = 2, x_2 = 7, y_2 = 10 \). Thus, the equation becomes \( D = \sqrt{(7 - 1)^2 + (10 - 2)^2} \).

Step 3: Calculating differences

Calculate the differences: \( 7 - 1 = 6 \) and \( 10 - 2 = 8 \).

Step 4: Squaring the differences

Square each difference: \((7 - 1)^2 = 6^2 = 36\) and \((10 - 2)^2 = 8^2 = 64\).

Step 5: Adding squares and taking square root

Add the squared differences: \(36 + 64 = 100\). Take the square root of this sum to find the distance: \(\sqrt{100} = 10\).

Key Concepts

Understanding the Coordinate Plane

The coordinate plane is an essential tool for locating and visualizing points in space. It consists of two perpendicular lines called axes: the horizontal line is the x-axis and the vertical line is the y-axis. Where they intersect is known as the origin, labeled as the point

• The x-axis measures the horizontal distance from the origin.
• The y-axis measures the vertical distance from the origin.

Each point on the coordinate plane is defined by a pair of numbers, called coordinates, written as

• The first number (x) indicates how far along the point is horizontally from the origin.
• The second number (y) shows how far up or down the point is vertically from the origin.

By using the coordinate plane, you can visually represent mathematical problems and calculate distances between points, as we can see in the given exercise.

Calculating Distance Between Points

Calculating the distance between two points on a coordinate plane is made simple using the distance formula. This formula is derived from the Pythagorean theorem. To calculate the distance

• Identify the coordinates of your points. For example: point
• Use the distance formula:
  If the points are

Let's substitute our example points into the distance formula. For points

• Compute the horizontal and vertical differences:
• Square both differences:
• Add these squared numbers together:
• Finally, take the square root of this sum to find the distance.

This process helps you determine the exact distance between two points quickly and accurately.

The Role of the Pythagorean Theorem

The Pythagorean theorem plays a crucial role when working on the coordinate plane and calculating distances. This theorem 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 other two sides. Mathematically, this is expressed as: \[c^2 = a^2 + b^2\]In the context of the coordinate plane:

• The horizontal and vertical differences between two points represent the two legs of a right triangle.
• The distance between the points is the hypotenuse of this triangle.
• The distance formula, \( D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), is a direct application of the Pythagorean theorem to find this hypotenuse.

By understanding and applying the Pythagorean theorem through the distance formula, you can easily calculate distances in various scenarios, making it a powerful tool in geometry and everyday math challenges.