To find the points of trisection of the line segment joining the points [tex]\( (0,0) \)[/tex] and [tex]\( (4,-4) \)[/tex], we follow these steps:
### Step 1: Identify the Coordinates of Point A and Point B
Let [tex]\( A = (0, 0) \)[/tex] and [tex]\( B = (4, -4) \)[/tex].
### Step 2: Calculate the Coordinates One-Third of the Way from Point A to Point B
We need to find the point that is located one-third of the way from [tex]\( A \)[/tex] to [tex]\( B \)[/tex]. The formula to find the point [tex]\( P \)[/tex] that divides the segment [tex]\( AB \)[/tex] in the ratio [tex]\( 1:2 \)[/tex] is given by:
[tex]\[
P = \left( \frac{2x_1 + x_2}{3}, \frac{2y_1 + y_2}{3} \right)
\][/tex]
Plugging in the coordinates for [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
P = \left( \frac{2 \cdot 0 + 4}{3}, \frac{2 \cdot 0 + (-4)}{3} \right) = \left( \frac{4}{3}, \frac{-4}{3} \right)
\][/tex]
So, the first point of trisection is [tex]\( \left( \frac{4}{3}, -\frac{4}{3} \right) \)[/tex].
### Step 3: Calculate the Coordinates Two-Thirds of the Way from Point A to Point B
Next, we need to find the point that is located two-thirds of the way from [tex]\( A \)[/tex] to [tex]\( B \)[/tex]. The formula to find the point [tex]\( Q \)[/tex] that divides the segment [tex]\( AB \)[/tex] in the ratio [tex]\( 2:1 \)[/tex] is given by:
[tex]\[
Q = \left( \frac{x_1 + 2x_2}{3}, \frac{y_1 + 2y_2}{3} \right)
\][/tex]
Plugging in the coordinates for [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
Q = \left( \frac{0 + 2 \cdot 4}{3}, \frac{0 + 2 \cdot (-4)}{3} \right) = \left( \frac{8}{3}, \frac{-8}{3} \right)
\][/tex]
So, the second point of trisection is [tex]\( \left( \frac{8}{3}, -\frac{8}{3} \right) \)[/tex].
### Step 4: Verify the Points
Let's write the results in decimal form to verify:
[tex]\[
\left( \frac{4}{3}, -\frac{4}{3} \right) \approx (1.3333, -1.3333)
\][/tex]
[tex]\[
\left( \frac{8}{3}, -\frac{8}{3} \right) \approx (2.6667, -2.6667)
\][/tex]
Therefore, the points of trisection of the line segment joining the points [tex]\( (0,0) \)[/tex] and [tex]\( (4,-4) \)[/tex] are:
[tex]\[
\left( 1.3333, -1.3333 \right) \quad \text{and} \quad \left( 2.6667, -2.6667 \right)
\][/tex]
