Problem 36 - Precalculus Mathematics for Calculus

Question

(a) Show that the points \((7,3)\) and \((3,7)\) are the same distance from the origin. (b) Show that the points \((a, b)\) and \((b, a)\) are the same distance from the origin.

Step-by-Step Solution

Answer

Both parts show that the points are equidistant from the origin using the distance formula.

Step 1: Calculate Distance for Point (7,3)

To find the distance from the origin to the point \((7,3)\), use the distance formula: \[ d = \sqrt{x^2 + y^2} \]Substitute \(x = 7\) and \(y = 3\):\[ d = \sqrt{7^2 + 3^2} = \sqrt{49 + 9} = \sqrt{58} \]

Step 2: Calculate Distance for Point (3,7)

Now calculate the distance from the origin to the point \((3,7)\) using the formula:\[ d = \sqrt{x^2 + y^2} \]Substitute \(x = 3\) and \(y = 7\):\[ d = \sqrt{3^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58} \]

Step 3: Compare Distances from Origin in Part (a)

The distance calculated for both points is \(\sqrt{58}\). Since they are equal, the points \((7,3)\) and \((3,7)\) are equidistant from the origin.

Step 4: Calculate Distance for Point (a,b)

Using the distance formula, calculate the distance from the origin to the point \((a, b)\): \[ d = \sqrt{a^2 + b^2} \]

Step 5: Calculate Distance for Point (b,a)

Use the same distance formula to calculate the distance from the origin to the point \((b, a)\):\[ d = \sqrt{b^2 + a^2} \]Since addition is commutative, \(a^2 + b^2 = b^2 + a^2\), both distances calculated are equivalent.

Step 6: Explain the Equality in Part (b)

Since both points \((a, b)\) and \((b, a)\) yield the same distance formula result, they are the same distance from the origin by the commutative property of addition.

Key Concepts

Understanding the Origin

The term "origin" in a coordinate plane refers to the point where the x-axis and the y-axis intersect. This point is denoted as \((0,0)\). It serves as a reference point from which we can measure distances of other points in the plane.
For example, when calculating the distance from the origin to a specific point like \((7,3)\) or \((3,7)\), we are measuring how far that point is positioned from this central axis intersection.

• The origin is crucial for applying the distance formula.
• It acts like a fixed anchor point in geometry.
• Distance from the origin helps in understanding how far away a point is located in the plane.

In any distance calculation using the distance formula, the origin plays a key role as the starting point for measuring distance.

The Concept of Equidistance

When we say two points are equidistant from the origin, it essentially means both points are exactly the same distance from the point \((0,0)\). This can be illustrated using the distance formula. For instance, when checking the distance of points \((7,3)\) and \((3,7)\) from the origin, both calculated distances come out to be \(\sqrt{58}\). Since they have equal distances, these points are equidistant.
Here’s what makes equidistance interesting:

• It helps to show symmetry or balance in geometrical setups.
• Equidistant points around the origin can vary in direction but equal in length of journey back to \((0,0)\).
• This property is useful for verifying equality of distance in geometric figures.

Equidistance serves to verify or illustrate relationships between geometric points by equal measure from a common location.

The Role of the Commutative Property in Distance

In mathematics, the commutative property applies to addition, stating that changing the order of numbers doesn’t change the sum. When calculating the distance for points \((a,b)\) and \((b,a)\), the property confirms that: \( a^2 + b^2 = b^2 + a^2\)Because addition is commutative, the result of the distance formula remains unchanged regardless of order.
This property is very useful:

• It simplifies calculations as order becomes irrelevant for sum.
• Ensures that point calculations like distances remain constant with varied variable positions.
• Assists in verifying distances by simplifying expressions.

This means even if points may appear different, their structured relationship with respect to the origin remains fully intact due to the commutative nature of addition.