Problem 11 - Algebra 1: Concepts and Skills

Question

Find the midpoint of the line segment with the given endpoints. Then show that the midpoint is the same distance from each given point. \((2,6),(4,2)\)

Step-by-Step Solution

Answer

The midpoint of the line segment connecting the points (2,6) and (4,2) is (3,4). The distance from each of these points to the midpoint is \(\sqrt{5}\), so the midpoint is the same distance from each point.

Step 1: Calculate the Midpoint

To find the midpoint of the line segment with the endpoints (2,6) and (4,2), the coordinates of the midpoint \(M\) are given by \((x_1 + x_2)/2 , (y_1 + y_2)/2\). Plugging the given points into the formula gives us \(M = ((2 + 4)/2 , (6 + 2)/2 ) = (3, 4)\).

Step 2: Calculate Distances

Next, calculate the distance from each point to the midpoint. The formula for the distance \(D\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). So, applying the distance formula between \(M\) and \((2,6)\) yields \(D_1 = \sqrt{(3 - 2)^2 + (4 - 6)^2} = \sqrt{5}\). Similarly, the distance between \(M\) and \((4,2)\) is \(D_2 = \sqrt{(4 - 3)^2 + (2 - 4)^2} = \sqrt{5}\).

Step 3: Compare the Distances

As demonstrated, the distances from both given points to the midpoint are both \(\sqrt{5}\). This shows that the midpoint of a segment is the same distance from each endpoint, which proves that the initially calculated midpoint is indeed the correct one.

Key Concepts

Midpoint Formula

Understanding the midpoint formula is essential in coordinate geometry. It is a straightforward technique for finding the exact center point between two points on a coordinate plane. The formula follows a simple average of the x-coordinates and the y-coordinates of the endpoints.

To find the midpoint, simply add up the x-coordinates of the two endpoints and divide by two for the x-coordinate of the midpoint. Repeat the process for the y-coordinates. Mathematically, if you have points \(A(x_1, y_1)\) and \(B(x_2, y_2)\), the midpoint \(M\) is obtained using the formula: \[(x_1 + x_2)/2 , (y_1 + y_2)/2\].

In the exercise, the midpoint of the line segment with endpoints \(A(2,6)\) and \(B(4,2)\) is found by averaging the x-coordinates (2 and 4) to get 3, and the y-coordinates (6 and 2) to get 4, resulting in midpoint \(M(3, 4)\). This midpoint divides the segment into two equal parts, equidistant from the endpoints.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is where algebra and geometry meet. It allows us to solve geometric problems using algebraic equations. Every point in the plane is represented by a pair of coordinates: the x-coordinate tells us how far along the point is, while the y-coordinate tells us how far up.

In coordinate geometry, shapes such as line segments, triangles, and circles are defined by equations or by a set of points. Concepts like slope, midpoint, and the distance between two points can be described using coordinates and formulas. The beauty of this approach is that it provides a clear method for calculating distances and other geometric properties directly from the coordinates of the points involved.

For instance, in our exercise, we used the given endpoints' coordinates to find the line segment's midpoint, a typical application of coordinate geometry.

Distance Formula

The distance formula is a fundamental tool in coordinate geometry that determines the distance between two points in the coordinate plane. It stems from the Pythagorean theorem in a right-angled triangle and reflects the idea of finding the hypotenuse length, which in this context is the straight-line distance between two points.

The formula is given by: \[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]. To use it, simply subtract the coordinates of the first point from the coordinates of the second point, square the results, add them together, and take the square root of the sum.

Application of this formula allowed us to verify that the distances from the endpoints \(A(2,6)\) and \(B(4,2)\) to the midpoint \(M(3, 4)\) are equal, both \(\sqrt{5}\), confirming the accuracy of the midpoint calculation. This is an essential proof that often accompanies problems in coordinate geometry, ensuring that a found midpoint is equidistant from each endpoint of the segment.