Question

Find the midpoint between the two points \((0,0),(0,8)\)

Short answer

The midpoint between the two points (0,0) and (0,8) is (0,4)

Step-by-step solution

Identify the coordinates of the two points

The two points given are (0,0) and (0,8). Here, for the first point (0,0), x1 = 0 and y1 = 0. For the second point (0,8), x2 = 0 and y2 = 8.

Apply the midpoint formula

The midpoint between any two points (x1, y1) and (x2, y2) can be found by using the formula: ((x1 + x2)/2 , (y1 + y2)/2). Therefore to find the midpoint, add the x-coordinates of the two points and divide by 2 and do the same for the y-coordinates.

Calculate the midpoint

By substituting the x and y coordinates into the formula, we get: Midpoint = ((0+0)/2 , (0+8)/2) = (0,4)

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a powerful bridge connecting algebra with geometry through the use of an XY coordinate system. It allows us to solve geometric problems using algebraic equations. For instance, the exercise at hand involves finding the midpoint between two points on a plane, which are identified by their coordinates.

These coordinates are pairs of numbers that represent positions relative to two reference lines, the X-axis (horizontal) and the Y-axis (vertical). By plotting points on this plane, we can visualize the problem and use algebraic methods to solve it. In the midpoint problem, we are essentially looking for a point that is equidistant from two given points, which is a common type of problem in coordinate geometry.

Algebraic Formulae

Algebraic formulae are the tools that allow us to express mathematical relationships and solve problems in a structured way. The midpoint formula \( \frac{x1 + x2}{2} , \frac{y1 + y2}{2} \) is one such tool in coordinate geometry that helps us compute the midpoint of a line segment efficiently.

Understanding and memorizing these formulae empower students to approach various problems methodically. When you apply the midpoint formula, you are performing algebraic operations—addition and division—on the coordinates of the endpoints of the segment. It is vital to substitute the correct values of \( x1, y1 \) and \( x2, y2 \) into the formula to obtain the correct midpoint. Algebraic formulae like this are essential for transforming geometric descriptions into calculated, exact locations on the coordinate plane.

Geometry Problem Solving

Geometry problem solving involves a systematic approach to tackling problems involving shapes, sizes, and the properties of space. When you approach a problem like finding the midpoint, it's crucial to first understand what the problem is asking. In this case, the problem is seeking a point that divides the segment into two equal lengths.

After comprehending the task, the next step is to gather the necessary information, which in our exercise means identifying the coordinates of the endpoints. Then one can apply the appropriate formula or theorem—in our case, the midpoint formula—to find the solution. Problem-solving in geometry often involves visual representation as well, and drawing a diagram can be incredibly helpful. With problems that involve more complex shapes or multiple steps, breaking down the problem into smaller, more manageable parts is also a recommended strategy.