Question

Find the coordinates of the points of trisection of the line segment joining the points \( A(-1, 4) \) and \( B(-3, -2) \).

Hint:

The trisection points are also the midpoints of segments formed by other trisection points. For example, Q is the midpoint of PB.

Answer 1

Given:
Points: \( A(-1, 4) \) and \( B(-3, -2) \)

We must find the two points that divide AB into three equal parts (trisection). These points divide AB internally in the ratios:
• 1 : 2 • 2 : 1

Let the points be P and Q.

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Point P divides AB in the ratio \( 1 : 2 \)
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Formula for internal division:
\[ P(x, y) = \left( \frac{m_2 x_1 + m_1 x_2}{m_1 + m_2},\ \frac{m_2 y_1 + m_1 y_2}{m_1 + m_2} \right) \] Here: \( m_1 = 1,\ m_2 = 2 \)
\( A(-1, 4),\ B(-3, -2) \)

Compute x-coordinate:
\[ x = \frac{2(-1) + 1(-3)}{3} = \frac{-2 - 3}{3} = -\frac{5}{3} \] Compute y-coordinate:
\[ y = \frac{2(4) + 1(-2)}{3} = \frac{8 - 2}{3} = \frac{6}{3} = 2 \] So, \[ P\left(-\frac{5}{3},\ 2\right) \]

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Point Q divides AB in the ratio \( 2 : 1 \)
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Now \( m_1 = 2,\ m_2 = 1 \)

Compute x-coordinate:
\[ x = \frac{1(-1) + 2(-3)}{3} = \frac{-1 - 6}{3} = -\frac{7}{3} \] Compute y-coordinate:
\[ y = \frac{1(4) + 2(-2)}{3} = \frac{4 - 4}{3} = 0 \] So, \[ Q\left(-\frac{7}{3},\ 0\right) \]

Final Answer:
The points of trisection of the line segment joining A and B are:

\[ \boxed{P\left(-\frac{5}{3},\ 2\right),\quad Q\left(-\frac{7}{3},\ 0\right)} \]