# Over or Under Determined?

Geometry Level 4

If $$\overline{US}=5$$ and $$\overline{UR}=4$$, find $$|R|$$

Details and assumptions:

$$|R|$$ means the magnitude of point $$R$$ from the origin.

(You really don't need to read everything below, it's simply a clarification of the diagram above).

Note: ignore points C and D on the diagram, they're incorrectly labeled.

Two different parallel lines $$A$$ and $$B$$ are drawn with the same slope $$m$$ such that $$m$$ is a positive integer, line $$A$$ intersects the origin, and line $$A$$ intersects the y-axis above line $$B$$.

A third line $$C$$ is drawn with slope $$n$$ such that $$0<n<m$$, it intersects line $$B$$ at point $$S$$ with coordinates $$(j,k)$$, and it also intersects the origin.

K, at this point you should probably just look at the diagram, but feel free to keep reading.

A forth line $$D$$ is drawn with slope $$p$$ such that $$p<0$$, it intersects line $$A$$ at point $$T$$ with coordinates $$(g,h)$$ such that $$g<j$$ and $$h<k$$, it intersects line $$C$$ at point $$R$$, and it has the same x intercept as line $$B$$.

A fifth line $$E$$ is drawn parallel to the x-axis such that point $$T$$ lies on this line and line $$E$$ intersects line $$C$$ at point $$U$$.

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