Question

The coordinates of the endpoints of \(\overline{G H},\) in the standard \((x, y)\) coordinate plane, are \((-8,-3)\) and \((2,3) .\) What is the \(x\) -coordinate of the midpoint of \(\overline{G H}\) ? F. \(-6\) G. \(-3\) H. 0 J. 3 K. 5

Short answer

Answer: G

Step-by-step solution

Identify the given coordinates

The coordinates of point G are (-8, -3), and the coordinates of point H are (2, 3).

Apply the midpoint formula for the x-coordinate

We are only interested in finding the x-coordinate of the midpoint. We use the formula: \(\frac{x_1 + x_2}{2}\), where \(x_1\) is the x-coordinate of point G and \(x_2\) is the x-coordinate of point H. \(x_m = \frac{-8 + 2}{2}\)

Calculate the value of the x-coordinate

Now, we simply need to solve for \(x_m\) in the expression: \(x_m = \frac{-8 + 2}{2} = \frac{-6}{2} = -3\) The x-coordinate of the midpoint of \(\overline{GH}\) is -3, which corresponds to answer choice G.

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves the study of geometric figures using a coordinate system. This approach allows for the representation of points, lines, and shapes within the context of two-dimensional plane. The standard coordinate system used in coordinate geometry is known as the Cartesian coordinate system.

In the context of the given exercise, we apply the principles of coordinate geometry to find the midpoint of a line segment. To do this, we plot the endpoints of the segment on the standard coordinate plane, which has a horizontal x-axis and a vertical y-axis. By identifying the coordinates of these endpoints, we engage in an application of coordinate geometry to solve a practical problem.

Standard Coordinate Plane

The standard coordinate plane is a two-dimensional surface consisting of two number lines that are perpendicular at their zero points, known as the origin. The horizontal number line is called the x-axis, and the vertical number line is the y-axis. Together, they form a grid where the position of any point can be expressed by an ordered pair of numbers, referred to as coordinates.

For instance, in our exercise, the endpoints of segment \(\overline{GH}\) are given as the ordered pairs \( (-8, -3) \) and \( (2, 3) \). The first number of each ordered pair represents the position on the x-axis, while the second indicates the position on the y-axis. By plotting these points on the standard coordinate plane, we visualize the line segment \(\overline{GH}\) and can further apply the midpoint formula.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. They are used to describe relationships between quantities and to solve equations. Algebraic expressions become particularly useful in describing formulas and solving problems in geometry.

The midpoint formula is a common algebraic expression that involves both arithmetic and variables. In our exercise, we used the midpoint formula for the x-coordinate: \( x_m = \frac{x_1 + x_2}{2} \), where \( x_1 \) and \( x_2 \) are x-coordinates of the endpoints of the line segment. We substituted the given numbers into the expression and simplified it to find the x-coordinate of the midpoint, which is a clear example of how algebraic expressions underpin the mechanics of coordinate geometry.