Problem 36 - Algebra and Trigonometry

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 pairs of points are the same distance from the origin.

Step 1: Understanding the Distance Formula

The distance formula to find the distance of a point (x, y) from the origin (0, 0) is given by:\[D = \sqrt{x^2 + y^2}\]This formula derives from the Pythagorean theorem applied to the right triangle formed by the point and the coordinate axes.

Step 2: Calculating Distance for (7,3)

Using the distance formula, calculate the distance from the point (7, 3) to the origin:\[D = \sqrt{7^2 + 3^2} = \sqrt{49 + 9} = \sqrt{58}\]So, the distance from the origin to the point (7, 3) is \(\sqrt{58}\).

Step 3: Calculating Distance for (3,7)

Now, calculate the distance from the point (3, 7) to the origin:\[D = \sqrt{3^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58}\]Hence, the distance from the origin to the point (3, 7) is also \(\sqrt{58}\).

Step 4: Conclusion for part (a)

Since both points (7, 3) and (3, 7) have the same distance \(\sqrt{58}\) from the origin, they are equidistant from the origin.

Step 5: Generalizing with Points (a,b) and (b,a)

Apply the distance formula to the point (a, b):\[D_1 = \sqrt{a^2 + b^2}\]Now apply the distance formula to the point (b, a):\[D_2 = \sqrt{b^2 + a^2}\]Both expressions \(D_1\) and \(D_2\) simplify to the same result: \(\sqrt{a^2 + b^2}\).

Step 6: Conclusion for part (b)

Thus, for any real numbers \(a\) and \(b\), the points (a, b) and (b, a) are always the same distance from the origin, \(\sqrt{a^2 + b^2}\).

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle of geometry that relates the sides of a right triangle. It states that in any 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 lengths of the other two sides. This can be expressed as \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.

In coordinate geometry, this concept is crucial because it helps us figure out the distances between points on a plane. By treating the line connecting two points as the hypotenuse of a right triangle, we can apply the Pythagorean theorem to find the exact distance. This is the foundation of the distance formula, which calculates the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a coordinate plane:

• The distance formula: \(D = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)
• This formula directly stems from \(a^2 + b^2 = c^2\)

This relationship shows how a simple geometric principle can have wide applications, including finding how far apart two locations are on a map.

Coordinate geometry

Coordinate geometry, also known as analytic geometry, merges algebra with geometry to describe and analyze the position of points, lines, and shapes on a plane. Points are specified by their coordinates, such as \((x, y)\), which can be plotted on a two-dimensional grid of horizontal (x-axis) and vertical (y-axis) lines.

This field provides the framework for using algebra to solve geometric problems, like calculating distances or slopes and proving concepts such as parallel lines or midpoints. The distance formula discussed in the exercise is a classic example of how coordinate geometry works.

• Coordinate geometry helps us translate geometric figures into algebraic equations
• It provides tools for calculating properties, such as area and perimeter, using coordinates
• Allows solving complex geometric problems through algebraic methods

By understanding these coordinates, students can more easily comprehend spatial relationships and motion, making it a crucial aspect of math education.

Equidistant points

Equidistant points are points that are the same distance apart from a particular point or line. In the context of the exercise, two points, such as \((7, 3)\) and \((3, 7)\), are equidistant from the origin if their calculated distance from it is the same.

Finding equidistant points involves understanding and applying the distance formula. In our example:

• Both \((7, 3)\) and \((3, 7)\) produce a distance of \(\sqrt{58}\) from the origin.
• Generalizing, you can demonstrate that any two points \((a, b)\) and \((b, a)\) are equidistant from the origin: \(\sqrt{a^2 + b^2}\).

Recognizing equidistant points helps in various mathematical problems and proofs, such as constructing certain shapes or proving congruence. This concept is not only beneficial in theoretical math but also practical in fields like design and architecture, where symmetry and balanced placement are required.