Question:medium

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

Show Hint

Once you have calculated the coordinates of the first point \(P\), you can find \(Q\) much faster by recognizing that \(Q\) is simply the midpoint of the line segment \(PB\):
\[ x_Q = \frac{x_P + x_B}{2} = \frac{-\frac{5}{3} - 3}{2} = \frac{-\frac{14}{3}}{2} = -\frac{7}{3} \]
\[ y_Q = \frac{y_P + y_B}{2} = \frac{2 + (-2)}{2} = 0 \]
Using the midpoint formula for the second point is easier and reduces the chance of manual error!
Updated On: Jul 7, 2026
  • \(\left(-\frac{5}{3}, 2\right)\) and \(\left(-\frac{7}{3}, 0\right)\)
  • \(\left(-\frac{4}{3}, 2\right)\) and \(\left(-\frac{5}{3}, 1\right)\)
  • \(\left(-\frac{5}{3}, -2\right)\) and \(\left(-\frac{7}{3}, 1\right)\)
  • (1, 2) and (2, 0)
Show Solution

The Correct Option is A

Solution and Explanation

Step 1: Find the first trisection point $P$ using the section formula with ratio $1:2$.
For $A(-1, 4)$ and $B(-3, -2)$, point $P$ divides $AB$ so that $AP:PB = 1:2$.
\[ x_P = \frac{1(-3) + 2(-1)}{1+2} = \frac{-3-2}{3} = -\frac{5}{3} \]
\[ y_P = \frac{1(-2) + 2(4)}{1+2} = \frac{-2+8}{3} = 2 \]
So $P = \left(-\frac{5}{3}, 2\right)$.

Step 2: Get the second trisection point $Q$ as the midpoint of $P$ and $B$, instead of applying the section formula a second time.
Since $AP = PQ = QB$, the point $Q$ sits exactly halfway between $P$ and $B$. The midpoint formula gives:
\[ x_Q = \frac{x_P + x_B}{2} = \frac{-\frac{5}{3} + (-3)}{2} = \frac{-\frac{5}{3} - \frac{9}{3}}{2} = \frac{-\frac{14}{3}}{2} = -\frac{7}{3} \]
\[ y_Q = \frac{y_P + y_B}{2} = \frac{2 + (-2)}{2} = \frac{0}{2} = 0 \]

Step 3: State the coordinates of $Q$.
\[ Q = \left(-\frac{7}{3}, 0\right) \]

Step 4: Confirm the pattern is consistent.
$P$ is one third of the way from $A$ to $B$, and $Q$, being the midpoint of $P$ and $B$, automatically lands at two thirds of the way from $A$ to $B$, exactly what a trisection point should be.

Final Answer:
The points of trisection are $\left(-\frac{5}{3}, 2\right)$ and $\left(-\frac{7}{3}, 0\right)$, which corresponds to option (A). \[ \boxed{\left(-\frac{5}{3}, 2\right) \text{ and } \left(-\frac{7}{3}, 0\right)} \]
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