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Chapter 1: Problem 49

Use the Midpoint Formula three times to find the three points that divide the line segment joining \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) into four parts.

Short Answer

Expert verified
The three points that divide the line segment into four equal parts are \(\left( \frac{x_1 + M_x}{2}, \frac{y_1 + M_y}{2}\right)\), \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\) and \(\left( \frac{M_x + x_2}{2}, \frac{M_y + y_2}{2}\right)\)

Step by step solution

01

Calculate Midpoint

First, find the midpoint M of the line segment joining \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) using midpoint formula. This will divide the line into two equal parts. The coordinates of M will be \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)
02

Calculate First Quarter Point

Now, find the midpoint point P of the line connecting \(\left(x_{1}, y_{1}\right)\) and M. This point will be one of the points that divides the line into four equal parts. So, the coordinates of P can be found with \(\left( \frac{x_1 + M_x}{2}, \frac{y_1 + M_y}{2}\right)\)
03

Calculate Third Quarter Point

Next, find the midpoint point Q of the line connecting M and \(\left(x_{2}, y_{2}\right)\). This will be the third point that divides the line into quarters. Q's coordinates can be calculated as \(\left( \frac{M_x + x_2}{2}, \frac{M_y + y_2}{2}\right)\)

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