Problem 19 - Algebra 1: Concepts and Skills

Question

Find the distance between the two points. Round your solution to the nearest hundredth if necessary. $$ (7,12),(-7,-4) $$

Step-by-Step Solution

Answer

The distance between the two points (7,12) and (-7,-4) is approximately 21.26.

Step 1: Write down the coordinates

We have two points, which we will call Point A and Point B. Point A has coordinates (7,12) and Point B has coordinates (-7,-4). In general, a point (x,y) in two dimensions will have x as the displacement along the x-axis (horizontal) and y as the displacement along the y-axis (vertical).

Step 2: Use the distance formula

The formula to find the distance between two points A(x1, y1) and B(x2, y2) is given by: \[d = \sqrt {(x2-x1)^2 + (y2-y1)^2}\] Substitute the coordinates of Point A into \(x1\) and \(y1\) and the coordinates of Point B into \(x2\) and \(y2\) respectively.

Step 3: Perform the substitution

Substituting 7 for \(x1\), 12 for \(y1\), -7 for \(x2\), and -4 for \(y2\), the formula becomes: \[d = \sqrt { (-7-7)^2 + (-4-12)^2}\]

Step 4: Perform the calculation

Solving the formula yields: \[d = \sqrt {(-14)^2 + (-16)^2}\] or \[d = \sqrt {196 + 256}\] which simplifies to \[d = \sqrt {452}\]

Step 5: Round to the nearest hundredth

Finally, to finish, take the square root of 452 and round to the nearest hundredth. This results in approximately 21.26.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses a coordinate system to study geometry. In this system, each point in the plane is represented by an ordered pair of numbers, called coordinates. These coordinates define the position of the point.

For example, a point labeled as \((x, y)\) signifies its location as:

• \(x\) - displacement along the horizontal x-axis
• \(y\) - displacement along the vertical y-axis

In our exercise, Point A is \((7, 12)\) and Point B is \((-7, -4)\). These pairs tell us exactly where each point is on the Cartesian plane.

The brilliance of coordinate geometry is that it allows us to measure distances, angles, and other geometric concepts using algebraic formulas. This adds a new layer of analytical depth to traditional geometry which relies solely on visual and spatial reasoning.

Distance Between Two Points

The distance between two points in a plane is a measure of the length of the line segment connecting them. When these points are represented in coordinate geometry, we can calculate this distance using a specific formula.

The distance formula is derived from the Pythagorean theorem and is given by:

• \[d = \sqrt {(x_2-x_1)^2 + (y_2-y_1)^2}\]

Here,

• \(d\) represents the distance
• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points

In our case, substituting in the coordinates of Point A \((7, 12)\) and Point B \((-7, -4)\), the formula becomes: \[d = \sqrt {(-7-7)^2 + (-4-12)^2}\]Solving this equation gives us the direct distance between the two points.

Distance is essential as it provides a clear-cut measure of separation, enabling us to quantify and compare how far apart different points lie on a coordinate plane.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, primarily applied to right-angled triangles. It is mathematically expressed as:

• \[a^2 + b^2 = c^2\]

where:

• \(a\) and \(b\) are the legs of the triangle
• \(c\) is the hypotenuse, the side opposite the right angle

In coordinate geometry, the Pythagorean theorem helps us establish the distance formula. By treating the difference in x-coordinates \((x_2 - x_1)\) and y-coordinates \((y_2 - y_1)\) as the two legs \(a\) and \(b\) of a right triangle, we find that the distance \(d\) between the points corresponds to the hypotenuse \(c\).

This relationship means that to find the direct distance between two points on a grid, we create an imaginary right-angled triangle with the line connecting the points as the hypotenuse. Thus, the distance formula: \[d = \sqrt {(x_2-x_1)^2 + (y_2-y_1)^2}\]originates from this theorem, allowing us to seamlessly transition between algebraic and geometric perspectives.