Problem 33 - Precalculus Mathematics for Calculus

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: Define the Distance Formula

The distance between a point \(x, y\) and the origin \(0, 0\) can be found using the formula \( ext{Distance} = \sqrt{x^2 + y^2}\). This formula is derived using the Pythagorean theorem.

Step 2: Calculate Distance of Point A

Substitute the coordinates of point A, \(6, 7\), into the distance formula: \[\text{Distance from origin to A} = \sqrt{6^2 + 7^2} = \sqrt{36 + 49} = \sqrt{85}\]

Step 3: Calculate Distance of Point B

Substitute the coordinates of point B, \(-5, 8\), into the distance formula:\[\text{Distance from origin to B} = \sqrt{(-5)^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89}\]

Step 4: Compare the Distances

Compare the values \(\sqrt{85}\) and \(\sqrt{89}\). Since \(\sqrt{85} < \sqrt{89}\), point A is closer to the origin.

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. This theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it is expressed as:\[ c^2 = a^2 + b^2 \]where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides. This theorem is the foundation for the distance formula used in coordinate geometry. By transforming the geometric concept into an algebraic form, we can calculate distances between points on a plane.

Coordinate geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where we use algebraic equations to represent geometric figures. By placing points on the Cartesian plane, we are able to create a visual representation of algebraic equations.

• The x-axis is the horizontal line, and the y-axis is the vertical line intersecting at the origin, which is point \((0,0)\).
• Each point on this plane is represented by an ordered pair \((x, y)\).
• This allows for the easy calculation of distances, slopes, midpoints, and more.

Coordinate geometry is crucial for solving problems involving distances and angles between points, like the exercise you encountered.

Distance calculation

Distance calculation in coordinate geometry often involves the distance formula, which is derived from the Pythagorean theorem. When you want to find the distance from a point \((x, y)\) to the origin \((0, 0)\), you utilize the formula:\[ \text{Distance} = \sqrt{x^2 + y^2} \]This becomes:

• For point A \((6, 7)\), \(\sqrt{6^2 + 7^2} = \sqrt{85}\).
• For point B \((-5, 8)\), \(\sqrt{(-5)^2 + 8^2} = \sqrt{89}\).

By comparing these values, you determine which point is closer to the origin. In this problem, point A is closer because \(\sqrt{85} < \sqrt{89}\). The use of the distance formula simplifies the process of comparing distances without needing a visual scale.