Problem 33 - Algebra and Trigonometry

Question

Which of the points \(A(6,7)\) or \(B(-5,8)\) is closer to the origin?

Step-by-Step Solution

Answer

Point A is closer to the origin.

Step 1: Understand the Distance Formula

The distance from a point \((x, y)\) to the origin \((0, 0)\) can be calculated using the distance formula: \[ d = \sqrt{x^2 + y^2} \]. Our task is to use this formula to calculate the distances of both given points \(A(6,7)\) and \(B(-5,8)\) to the origin, and then compare these distances.

Step 2: Calculate Distance of Point A from Origin

Substitute \(x = 6\) and \(y = 7\) into the distance formula to find the distance of point \(A\) from the origin.\[ d_A = \sqrt{6^2 + 7^2} = \sqrt{36 + 49} = \sqrt{85} \]

Step 3: Calculate Distance of Point B from Origin

Substitute \(x = -5\) and \(y = 8\) into the distance formula to find the distance of point \(B\) from the origin.\[ d_B = \sqrt{(-5)^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89} \]

Step 4: Compare Distances

Now we compare \(d_A = \sqrt{85}\) and \(d_B = \sqrt{89}\). Since \(\sqrt{85} < \sqrt{89}\), point \(A\) is closer to the origin than point \(B\).

Key Concepts

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics allows us to graph points, lines, and figures on a two-dimensional plane using a set of coordinates. Each point on this plane is defined by a pair of numbers, usually written as \((x, y)\). The first number, \(x\), represents the horizontal position, while the second number, \(y\), indicates the vertical position.
By using a coordinate system, we can accurately describe geometric figures, calculate angles and distances, and solve complex problems involving shapes and figures. This makes coordinate geometry a powerful tool in fields ranging from physics to computer graphics. Essential concepts include understanding the coordinate plane itself, how to plot points, and how to interpret and manipulate equations that represent various geometric forms.
In our specific problem, the points \(A(6,7)\) and \(B(-5,8)\) exist on this coordinate plane. By understanding their positions using coordinate geometry, we can apply formulas and concepts to find distances and relationships between them.

Calculating Distances

To calculate distances on a coordinate plane, we use the distance formula. This formula derives from the Pythagorean theorem and calculates the distance between two points in a plane. When one of these points is the origin \((0,0)\), our calculation simplifies to the formula: \[ d = \sqrt{x^2 + y^2} \].
This formula allows us to determine the direct distance from any point \((x, y)\) to the origin. Here's how it works:

• First, square the \(x\) and \(y\) values. This removes any negative signs and provides a positive value indicative of each coordinate's distance from the axes.
• Next, add these squared values together. This step combines the horizontal and vertical distances into a single, cumulative measure.
• Finally, take the square root of this sum. The result is the direct distance from the specified point to the origin.

In our example, applying the distance formula to point \(A(6,7)\) gives \[ \sqrt{36 + 49} = \sqrt{85} \], while for point \(B(-5,8)\) it results in \[ \sqrt{25 + 64} = \sqrt{89} \]. These calculations reveal how far each point lies from the origin.

Comparing Distances

Once we have calculated distances from the origin to each point, our next step is to compare them. Comparing distances is often about understanding which is shorter or longer, which is essential in making decisions about positioning and proximity.
To compare the distances calculated in our problem: - Point \(A\) is at a distance of \(\sqrt{85}\) units from the origin.- Point \(B\) is \(\sqrt{89}\) units away.The values can be compared directly as they are both in square root form and positive. Since \(\sqrt{85} < \sqrt{89}\), we conclude that point \(A\) is closer to the origin than point \(B\).
This method of comparison helps determine relative positions not just in simple problems like ours, but in complex systems and applications involving multiple points and distances. Understanding which points are closer or farther apart can have practical applications in real-world scenarios, such as optimizing routes, planning spaces, or analyzing spatial data.