Problem 24 - Algebra 1: Concepts and Skills

Question

Find the midpoint of the line segment connecting the given points. Then show that the midpoint is the same distance from each point. \((7,-3),(-1,-9)\)

Step-by-Step Solution

Answer

The midpoint of the line segment connecting the points (7,-3) and (-1,-9) is (3, -6). The distance between each of the given points and this midpoint is 5 units.

Step 1: Finding the Midpoint

The first task is to use the formula for finding the midpoint: for points (x_1, y_1) and (x_2, y_2), the midpoint M is given by:\[ M = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right) \],Given points are (7,-3),(-1,-9). Inserting these coordinates yields:\[ M = \left( \frac{7 + (-1)}{2}, \frac{-3 + (-9)}{2} \right) = (3, -6)\]

Step 2: Verify the Midpoint

The next step is to confirm that the calculated midpoint is equidistant from each of the original points. This verification is done by calculating the distance from each point to the midpoint. The distance D between two points (x1, y1) and (x2, y2) is given by:\[ D = \sqrt{(x2-x1)^2 + (y2-y1)^2} \],So, the distance from the first point (7,-3) to the midpoint (3,-6) is:\[ D1 = \sqrt{(3-7)^2 + ((-6)-(-3))^2} = \sqrt{16 + 9} = \sqrt{25}= 5\],Similarly, the distance from the second point (-1,-9) to the midpoint (3,-6) is:\[ D2 = \sqrt{(3-(-1))^2 + ((-6)-(-9))^2} = \sqrt{16 + 9} = \sqrt{25}= 5\],Since D1 = D2, the midpoint is equidistant from both endpoints.

Key Concepts

Distance Formula

The distance formula is a mathematical tool used to find the distance between two points on a coordinate plane. It is derived from the Pythagorean theorem and gives a straightforward way to calculate how far apart two points are. If you have two points,

• the first being \((x_1, y_1)\) and
• the second \((x_2, y_2)\),

you can use the distance formula \[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]. This formula finds the length of the straight line connecting the two points. It does this by calculating the square root of the sum of the squares of the differences in the \(x\) and \(y\) coordinates.Using the distance formula is handy whenever you need to measure between points and verify distances in problems involving coordinate geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to handle geometric problems about positioning of points on a plane. It uses a coordinate system, typically a Cartesian plane, to describe the location of points through pairs of numbers, which are

• \((x, y)\) by convention.

In coordinate geometry, you can solve various geometric questions, such as finding distances between points, midpoints of line segments, and even the equations of lines. This approach is beneficial since it allows you to apply algebraic techniques to solve geometric problems. The coordinates simplify the process of calculating properties like distance and midpoint, making it easier to visualize relationships between different geometric figures.

Equidistant Points

Equidistant points are points that have the same distance from a particular point or line. In coordinate geometry, equidistant points come into play when verifying specific properties, such as whether a point is the midpoint of a line segment.When a point,

• like \((3, -6)\),

is calculated as the midpoint of the line segment formed by two points, \((7, -3)\) and \((-1, -9)\), it is essential to check that this midpoint is equidistant from both endpoints. By applying the distance formula to find the distance from the midpoint to each end, we can confirm the point is valid. Here, we have \(D1 = \sqrt{25} = 5\) and \(D2 = \sqrt{25} = 5\), meaning each distance is equal, verifying equidistance. In problems involving symmetry and specific geometric properties, identifying and confirming equidistant points is fundamental.