Show that the coordinates (x^', y^') and (x^'', y^'') of the points of trisection of the segment

step 1 We are trying to prove that the first trisection point occurs at a certain point that was listed

Show that the coordinates (x^', y^') and (x^'', y^'') of the...

Show that the coordinates $\left(x^{\prime}, y^{\prime}\right)$ and $\left(x^{\prime \prime}, y^{\prime \prime}\right)$ of the points of trisection of the segment with endpoints $\mathbf{S}\left(x_1, y_1\right)$ and $\mathbf{T}\left(x_2, y_2\right)$ are given by $$ x^{\prime}=\frac{2 x_1+x_2}{3}, \quad y^{\prime}=\frac{2 y_1+y_2}{3} $$ and $$ x^{\prime \prime}=\frac{x_1+2 x_2}{3}, \quad y^{\prime \prime}=\frac{y_1+2 y_2}{3} $$

Show that the coordinates $\left(x^{\prime}, y^{\prime}\right)$ and $\left(x^{\prime \prime}, y^{\prime \prime}\right)$ of the points of trisection of the segment with endpoints $\mathbf{S}\left(x_1, y_1\right)$ and $\mathbf{T}\left(x_2, y_2\right)$ are given by

$$
x^{\prime}=\frac{2 x_1+x_2}{3}, \quad y^{\prime}=\frac{2 y_1+y_2}{3}
$$

and

$$
x^{\prime \prime}=\frac{x_1+2 x_2}{3}, \quad y^{\prime \prime}=\frac{y_1+2 y_2}{3}
$$

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Key Concepts

Section Formula

The section formula is a fundamental concept in coordinate geometry that provides a method to determine the coordinates of a point that divides a line segment connecting two points in a given ratio. It offers a systematic way of averaging the endpoints' coordinates weighted by their relative distances from the dividing point, which is essential in establishing the positions of points along a line.

Internal Division

Internal division refers to the scenario where a point lies between the endpoints of a line segment and divides the segment in a specific ratio. This concept is crucial for understanding how the distances from the two endpoints contribute to the exact position of the dividing point, as seen in the derivation of the trisection points.

Trisection

Trisection involves dividing a line segment into three equal parts. In coordinate geometry, this concept is applied by using the section formula to compute the coordinates of the points that partition the segment into three segments of equal length. The process highlights the use of weighted averages of the endpoint coordinates based on the specific division ratio.

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