Problem 10 - 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 \(A B\). $$ A(-2,-5), \quad B(4,6) $$

Step-by-Step Solution

Answer

Distance: \(\sqrt{157} \approx 12.53\); Midpoint: \((1, 0.5)\).

Step 1: Distance Formula

To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula:\[ d(A, B) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Step 2: Substitute the Points into the Distance Formula

The coordinates for point \(A\) are \((-2, -5)\) and for point \(B\) are \((4, 6)\). Substitute these into the distance formula:\[ d(A, B) = \sqrt{(4 - (-2))^2 + (6 - (-5))^2} \] \[ = \sqrt{(4 + 2)^2 + (6 + 5)^2} \]

Step 3: Calculate the Squared Differences

Calculate the squared differences:\[ d(A, B) = \sqrt{6^2 + 11^2} \] \[ = \sqrt{36 + 121} \]

Step 4: Calculate the Distance

Sum the squared differences and take the square root:\[ d(A, B) = \sqrt{157} \] \[ \approx 12.53 \]

Step 5: Midpoint Formula

To find the midpoint of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\), use the midpoint formula:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Step 6: Substitute the Points into the Midpoint Formula

Substitute the coordinates for points \(A\) and \(B\) into the midpoint formula:\[ M = \left( \frac{-2 + 4}{2}, \frac{-5 + 6}{2} \right) \] \[ = \left( \frac{2}{2}, \frac{1}{2} \right) \]

Step 7: Calculate the Midpoint

Complete the calculations to find the midpoint:\[ M = (1, 0.5) \]

Key Concepts

Midpoint Formula

The midpoint formula allows us to find the middle point of a line segment defined by two endpoints. It’s like finding that sweet spot right in the center of a line! Imagine your two endpoints are \((x_1, y_1)\) and \((x_2, y_2)\). The midpoint formula is: \[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]To use it, simply add the \(x\) coordinates together and divide by 2. Do the same with the \(y\) coordinates. This gives the exact half-way point between the two.Let’s try this with points \(A = (-2, -5)\) and \(B = (4, 6)\). Plugging their coordinates into the formula:- Add the \(x\) values: \((-2) + 4 = 2\), then divide by 2: \(\frac{2}{2} = 1\).- Add the \(y\) values: \((-5) + 6 = 1\), then divide by 2: \(\frac{1}{2} = 0.5\).So, the midpoint of \(AB\) is \((1, 0.5)\). This midpoint splits the line segment into two equal parts, each with distinct yet symmetrical halves.

Coordinates

Coordinates provide a way to pinpoint exact locations on the two-dimensional plane. Think of them as the geographical address for any point. Coordinates are usually expressed as a pair \((x, y)\).For instance, the point \((4, 6)\) where:- \(x = 4\) represents the horizontal movement from the origin - \(y = 6\) indicates the vertical movement from the originThese coordinates allow us to easily describe positions and perform geometric calculations, like finding distances or midpoints.When you encounter points like \((x_1, y_1)\) and \((x_2, y_2)\), they denote specific positions in this plane. Knowing how to retrieve and use coordinates ensures that you can assess relationships between points, such as determining how far apart they are or the middle point between them.

Distance Between Points

To measure how far apart two points are on a plane, utilize the distance formula, which is derived directly from the Pythagorean theorem. A visual way to understand this is to imagine the line segment between the two points as the hypotenuse of a right triangle.The distance formula is:\[d(A, B) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula calculates the direct line distance between two points by considering the change in \(x\) and \(y\) coordinates.For points \(A=(-2, -5)\) and \(B=(4, 6)\):

• Find the difference in \(x\) coordinates: \(4 - (-2) = 6\)
• Find the difference in \(y\) coordinates: \(6 - (-5) = 11\)
• Square each difference: \(6^2 = 36\) and \(11^2 = 121\)
• Sum the squares: \(36 + 121 = 157\)
• Take the square root: \(\sqrt{157} \approx 12.53\)

Through these steps, we find the distance between points \(A\) and \(B\) is approximately \(12.53\), reflecting the direct line segment connecting them. This concept helps in understanding spatial relationships in geometry and everyday life scenarios like mapping or navigation.