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# Help: Dividing line with ratio

A segment $$PQ$$ with $$2.1$$ in length divided by point $$X$$ so that $$\dfrac{PX}{XQ} = \dfrac{4}{3}$$ and by point $$Y$$ so that $$\dfrac{PY}{YQ} = -\dfrac{4}{3}$$. Find $$XY$$.

Note by Jason Chrysoprase
10 months ago

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We use the formulae for internal and external division of a line segment in a ratio:

If the division is $$\frac {m}{n}$$ of segment $$AB$$ and the internal point is $$C$$ and the external point is $$D$$, the internal length $$BC$$ is $$\frac {n}{m+n} AB$$ and the external length $$BD$$ is $$\frac {n}{m-n}$$.

Try to prove these formulae. We have

$$QX=\frac {3}{4+3} PQ \Rightarrow QX = 0.9$$

$$QY = \frac {3}{4-3} PQ \Rightarrow QY = 6.3$$

Thus, $$QX+QY=XY=7.2$$. · 10 months ago