Problem 19 - Contemporary Precalculus

Question

Find the distance between the two points and the midpoint of the segment joining them. $$(a, b),(b, a)$$

Step-by-Step Solution

Answer

Answer: The distance between the points is \(d = \sqrt{2} \cdot |b - a|\) and the midpoint is \(M = \left(\frac{a + b}{2}, \frac{a + b}{2}\right)\).

Step 1: Use the distance formula

The distance formula is given by: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Here, \((x_1, y_1) = (a, b)\) and \((x_2, y_2) = (b, a)\). Plug in these coordinates into the formula and simplify.

Step 2: Calculate the distance

Using the distance formula, we have: $$d = \sqrt{(b - a)^2 + (a - b)^2}$$ Now, simplify the expression: $$d = \sqrt{2(b - a)^2}$$ Then, take the square root of 2: $$d = \sqrt{2} \cdot |b - a|$$

Step 3: Use the midpoint formula

Now, we need to find the midpoint of the line segment. The midpoint formula is given by: $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$ Here, \((x_1, y_1) = (a, b)\) and \((x_2, y_2) = (b, a)\). Plug in these coordinates into the formula.

Step 4: Calculate the midpoint

Using the midpoint formula, we have: $$M = \left(\frac{a + b}{2}, \frac{b + a}{2}\right)$$ Now, simplify the expression: $$M = \left(\frac{a + b}{2}, \frac{a + b}{2}\right)$$ So, the distance between the two points is \(d = \sqrt{2} \cdot |b - a|\) and the midpoint of the line segment joining them is \(M = \left(\frac{a + b}{2}, \frac{a + b}{2}\right)\).

Key Concepts

Midpoint Formula

The midpoint formula is a simple yet essential concept in geometry. It helps us find the exact middle point between two coordinates on a plane. This can be useful in various applications, such as dividing a line segment equally or determining symmetry. The formula is:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

• Coordinates: For two points \((x_1, y_1)\) and \((x_2, y_2)\), the formula takes the average of the x-coordinates and the average of the y-coordinates.
• Symmetry: The midpoint gives us a point that is equidistant from both endpoints of the segment.
• Example: Using the points \((a, b)\) and \((b, a)\), the midpoint is \(M = \left( \frac{a + b}{2}, \frac{a + b}{2} \right)\). This shows the midpoint lies exactly halfway along both axes.

Understanding how to correctly apply the midpoint formula can simplify many problems in geometry and other fields. It makes dividing and analyzing data more efficient and precise.

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, is a method of representing geometric shapes in numerical form. It combines algebra with geometry to explore relationships between points, lines, and curves on the coordinate plane.

• Basic Elements: Points, lines, and shapes are defined using coordinates on an axis, like \((x, y)\).
• Equations: Lines and curves can be represented using algebraic equations, such as linear equations for lines.
• Applications: Coordinate geometry is used in various fields like engineering, architecture, and computer graphics, where precise measurement and calculation are crucial.

By leveraging coordinate geometry, complex geometric problems can be solved with algebraic rules and formulas, making it a powerful tool for both abstract mathematical exploration and practical problem-solving.

Analytic Geometry

Analytic geometry is a fascinating branch of mathematics that focuses on solving geometric problems using algebraic equations and methods. It involves the real number system and a coordinate plane to precisely measure and analyze different shapes.

• Intersection: Analytic geometry helps find where geometric shapes intersect, such as where lines cross or circles touch.
• Distance and Midpoints: With formulas like the distance and midpoint formulas, we can calculate exact measurements between points.
• Equation-solving: Problems that might seem complex geometrically can often be simplified into algebraic equations.

While classical geometry deals with shapes and their properties in an abstract way, analytic geometry provides a way to understand and work with these by using coordinates and equations. It's a bridge that connects numeric calculations to spatial reasoning, making it highly versatile and insightful.