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Problem 2

# The line joining the points $$(1,-2)$$ and $$(-3,4)$$ is trisected; find the co- ordinates of the points of trisection.

Expert verified
The coordinates of the points of trisection are $$(-1,2)$$ and $$(-1,0)$$.
See the step by step solution

## Step 1: Identify the coordinates of the points

The coordinates of the points are $$(1,-2)$$ and $$(-3,4)$$. We can label them as $$A(1,-2)$$ and $$B(-3,4)$$ for clarity.

## Step 2: Calculate the coordinates of the first point of trisection

We can find the coordinates of the first point of trisection (suppose $$P$$) using the section formula, which when a point divides a line internally in the ratio $$m:n$$ is given by $\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)$ where $$(x_1,y_1)$$ and $$(x_2,y_2)$$ are the coordinates of the two points, and $$m$$ and $$n$$ are the two parts in which the line is divided. Here, the ratio is $$2:1$$ with $$m=2$$ and $$n=1$$, and the points are $$(1,-2)$$ and $$(-3, 4)$$. Substituting these values, we get the first point of trisection as $$P\left(\frac{2*(-3) + 1*1}{2+1}, \frac{2*4 +(-2)*1}{2+1}\right)$$ leading to coordinates $$(-1,2)$$.

## Step 3: Calculate the coordinates of the second point of trisection

We can find the coordinates of the second point of trisection (suppose $$Q$$) in a similar way as step 2, only this time the ratio is $$1:2$$ with $$m=1$$ and $$n=2$$, and the points are still $$(1,-2)$$ and $$(-3, 4)$$. By substituting these values, we get the second point of trisection to be $$Q\left(\frac{1*(-3) + 2*1}{1+2}, \frac{1*4 +(-2)*2}{1+2}\right)$$ leading to coordinates $$(-1,0)$$.

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