Author Jonathan David
Author Jonathan David
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Use the Midpoint Formula

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Using the Midpoint Formula in Trigonometry

Introduction

The midpoint formula is a simple yet powerful tool used to find the point that lies exactly halfway between two given points in a Cartesian coordinate system. This formula is essential for solving geometric problems and analyzing the properties of line segments in trigonometry.

The Midpoint Formula

The midpoint M of a line segment connecting two points (x_1, y_1) and (x_2, y_2) is given by:

    \[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]

This formula averages the x-coordinates and y-coordinates of the two points to find the coordinates of the midpoint.

Steps to Use the Midpoint Formula

Follow these steps to calculate the midpoint:

  1. Identify the coordinates of the two points as (x_1, y_1) and (x_2, y_2).
  2. Add the x-coordinates: x_1 + x_2.
  3. Add the y-coordinates: y_1 + y_2.
  4. Divide the sum of the x-coordinates by 2 to find the x-coordinate of the midpoint.
  5. Divide the sum of the y-coordinates by 2 to find the y-coordinate of the midpoint.

Example Problem

Find the midpoint of the points (3, 4) and (7, 8).

Step 1: Identify the Coordinates

(x_1, y_1) = (3, 4) and (x_2, y_2) = (7, 8)

Step 2: Apply the Formula

Use the midpoint formula:

    \[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]

Substitute the values:

    \[M = \left(\frac{3 + 7}{2}, \frac{4 + 8}{2}\right)\]

Step 3: Simplify

    \[M = \left(\frac{10}{2}, \frac{12}{2}\right)\]

    \[M = (5, 6)\]

The midpoint is (5, 6).

Practice Problems

Try solving these problems using the midpoint formula:

  1. Find the midpoint of (2, 3) and (10, 7).
  2. Calculate the midpoint of (-5, -4) and (1, 2).
  3. Prove that the midpoint of (0, 0) and (6, 8) lies on the line segment connecting the two points.

By mastering the midpoint formula, you can easily solve problems involving line segments and geometric constructions. Practice regularly to enhance your skills!

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