Problem 11 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 9-14: (a) Find the distance \(d(A, B)\) between \(A\) and \(B\). (b) Find the midpoint of the segment \(A B\). $$ A(-5,0), \quad B(-2,-2) $$

Step-by-Step Solution

Answer

Distance: \(\sqrt{13}\); Midpoint: \((-\frac{7}{2}, -1)\).

Step 1: Determine the Distance Formula

The distance formula to find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ d(A, B) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].

Step 2: Substitute the Coordinates of Points A and B into the Distance Formula

Given points \(A(-5,0)\) and \(B(-2,-2)\), substitute \(x_1 = -5\), \(y_1 = 0\), \(x_2 = -2\), and \(y_2 = -2\) into the distance formula: \[ d(A, B) = \sqrt{(-2 - (-5))^2 + (-2 - 0)^2} \].

Step 3: Calculate the Distance

Perform the calculations to find \(d(A, B)\):\[ d(A, B) = \sqrt{(3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \].

Step 4: Determine the Midpoint Formula

The midpoint formula for two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \].

Step 5: Substitute the Coordinates of Points A and B into the Midpoint Formula

Use the coordinates of \(A(-5,0)\) and \(B(-2,-2)\) and substitute them into the midpoint formula:\(x_1 = -5\), \(y_1 = 0\), \(x_2 = -2\), \(y_2 = -2\).\[ M = \left( \frac{-5 + (-2)}{2}, \frac{0 + (-2)}{2} \right) \].

Step 6: Calculate the Midpoint

Perform the calculations to find the midpoint \(M\):\[ M = \left( \frac{-7}{2}, \frac{-2}{2} \right) = \left( -\frac{7}{2}, -1 \right) \].

Key Concepts

Distance Formula

The Distance Formula is a handy tool in coordinate geometry, used to calculate the distance between two points on a plane. If you have two points, let's say \(A(x_1, y_1)\) and \(B(x_2, y_2)\), the distance between them is calculated using the formula: \[ d(A, B) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] This formula is derived from the Pythagorean theorem. It essentially finds the hypotenuse of a right-angled triangle formed by the difference in x-coordinates and y-coordinates of the points.

• Steps to use the Distance Formula:
• Find the change in x (\(x_2 - x_1\))
• Find the change in y (\(y_2 - y_1\))
• Square both changes
• Add them together
• Take the square root of the sum

By applying these steps, you can find the distance between any two points efficiently. For example, the distance between the points \((-5,0)\) and \((-2,-2)\) is \(\sqrt{13}\). It's as simple as plugging the coordinates into the formula and calculating.

Midpoint Formula

The Midpoint Formula is a method used to find the point that is exactly halfway between two points on a coordinate plane. This point is known as the midpoint. Suppose we have two endpoints of a line segment: \(A(x_1, y_1)\) and \(B(x_2, y_2)\). The midpoint \(M\) can be found using:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]This formula gives the average of the x-coordinates and the y-coordinates for the two endpoints.

• Steps to find the midpoint:
• Add the x-coordinates together, then divide by 2.
• Add the y-coordinates together, then divide by 2.

This method is quite effective for evenly splitting any line segment into two equal parts. For instance, using points \((-5,0)\) and \((-2,-2)\), once calculated, the midpoint is \((-\frac{7}{2}, -1)\), marking the exact middle point of the segment.

Coordinate Geometry

Coordinate Geometry, also known as analytic geometry, is a branch of mathematics that deals with points, lines, and shapes using coordinates. It provides a systematic way to analyze geometric shapes algebraically. This involves using the coordinate plane to provide visual representation and solutions to geometric problems.
Key Concepts in Coordinate Geometry:

• Coordinates: Specify the position of points using pairs of numbers (x, y).
• Distance: Calculate understanding using the distance formula and how far two points are from each other.
• Midpoint: Determine the center point of a line segment with the midpoint formula.
• Equations of Lines: Understand slopes and y-intercepts to form equations representing lines.

Coordinate geometry links algebra and geometry. This connection lets us solve problems by converting geometry shapes into algebraic equations. It's not only crucial in math, but it helps in fields like physics, engineering, and computer science where spatial understanding is vital.