Problem 15 - College Algebra Essentials

Question

find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (3 \sqrt{3}, \sqrt{5}) \text { and }(-\sqrt{3}, 4 \sqrt{5}) $$

Step-by-Step Solution

Answer

The distance between points (3*sqrt(3), sqrt(5)) and (-sqrt(3), 4*sqrt(5)) is approximately 9.64 units.

Step 1: Identify the coordinates of points

The coordinates of first point A are (3*sqrt(3), sqrt(5)) and that of point B are (-sqrt(3), 4*sqrt(5)).

Step 2: Use the formula of Distance

The Distance formula derived from the Pythagoras theorem will be used here. It is given by: Distance = sqrt[(x2 - x1)² + (y2 - y1)²]. Here, (x1, y1) are the coordinates of point A and (x2, y2) are the coordinates of point B.

Step 3: Substitute the coordinates into the formula

By using the formula, we get Distance = sqrt [(-sqrt(3) - 3*sqrt(3))² + (4*sqrt(5) - sqrt(5))²].

Step 4: Simplify the expression

On simplifying we get, Distance = sqrt [(-4*sqrt(3))² + (3*sqrt(5))²] = sqrt [48 + 45] = sqrt [93].

Step 5: Evaluate the square root

Evaluating the square root gives approximate value of Distance = 9.64 when rounded to two decimal places.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in mathematics that relates the lengths of the sides of a right-angled triangle. It states that the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. The theorem can be written as: \[ c^2 = a^2 + b^2 \] In the context of coordinate geometry, we apply this theorem to find the distance between two points in a plane by considering the difference in their coordinates as the two shorter sides of a right-angled triangle. - **Right-Angled Triangle:** Imagine a triangle formed by dropping vertical and horizontal lines from each point to form the missing side. - **Hypotenuse as Distance:** The distance between the two points acts as the hypotenuse. Using this method, we derive the distance formula, which helps calculate the straight-line distance between any two points in a 2D space.

Square Roots

Square roots are used to find a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \(3 \times 3 = 9\). In the distance formula, we take the square root after summing up the squares of differences. This step is crucial because it ensures we are finding the actual straight-line distance, not just the squared value.- **Notation:** The square root is usually denoted by the symbol \( \sqrt{} \). - **Calculation:** To use the square root in our exercise, calculate \( \sqrt{93} \) to find the precise distance between the points, and round it to two decimal places if needed. This is what transforms the calculated value into a number that explains the real-world "distance" on the plane. Practice calculating square roots to improve speed and accuracy, which is especially helpful in tests and assessments.

Coordinate Geometry

Coordinate Geometry, or Analytic Geometry, is the study of geometry using a coordinate system. By placing geometric figures in a coordinate plane, we can use algebraic formulas to describe these figures' attributes and solve geometric problems.- **Coordinates:** These are pairs of numbers that denote a specific location in a two-dimensional space, typically written as \((x, y)\). - **Distance Calculation:** In our exercise, the coordinates are given as \((3\sqrt{3}, \sqrt{5})\) and \((-\sqrt{3}, 4\sqrt{5})\). By applying the distance formula, we examine these values to get to the physical distance between the points.Coordinate geometry allows us to see how geometric properties and algebraic equations, such as the distance formula, work together. It is vital for students to label and apply these coordinates properly when solving related problems. This understanding will aid in solving more complex problems in both pure and applied mathematics.