Midpoints in the complex plane

Learning Objectives Represent complex numbers as position vectors in the Argand diagram. Derive and state the formula for the midpoint of a line segment connecting two complex numbers. Accurately calculate the midpoint of the line segment connecting two given complex numbers. Algebraically determine a missing endpoint when given one endpoint and the midpoint. Interpret the midpoint of two complex numbers as their arithmetic mean. Apply the midpoint formula to solve geometric problems in the complex plane, such as finding the center of a diagonal. If two ships are located at positions represented by complex numbers, where is the exact halfway point for a rescue helicopter to meet them? 🚁 This tutorial connects the familiar concept of a midpoint from coordinate geometry to t...

TermDefinitionExample Complex Plane (Argand Diagram)A two-dimensional plane used to represent complex numbers. The horizontal axis is the real axis (Re) and the vertical axis is the imaginary axis (Im).The complex number z = 3 + 4i is represented by the point (3, 4) on the complex plane. Position VectorIn the context of the complex plane, a vector originating from the origin (0 + 0i) and terminating at the point representing a complex number z = a + bi.The complex number z = -2 + i can be visualized as a position vector from the origin to the point (-2, 1). MidpointThe point on a line segment that is equidistant from both endpoints.The midpoint of the line segment between (0,0) and (10,0) is (5,0). Complex Number AdditionThe process of adding two complex numbers by summing their real and...

The Midpoint Formula z_m = \frac{z_1 + z_2}{2} To find the midpoint (z_m) of the line segment connecting two complex numbers (z₁ and z₂), add the complex numbers together and divide the result by 2. This is equivalent to averaging the two numbers. The Endpoint Formula z_2 = 2z_m - z_1 Use this formula to find a missing endpoint (z₂) when you are given the other endpoint (z₁) and the midpoint (z_m). It is derived by rearranging the Midpoint Formula.