Question

UM=2x+14
MS=14+x
Find US

Answer 1

Answer on Question #72400 – Math – Geometry

Question

$$UM = 2x + 14$$ $$MS = 14 + x$$

Find US

Solution

In the common case, the given points $U$, $M$ and $S$ (the end points of $UM$, $MS$ and $US$) form a triangle $UMS$ (see Fig. 1). We can apply the cosine rule to find unknown side $US$:

$$US^2 = UM^2 + MS^2 - 2UM \cdot MS \cos \alpha,$$

where $\alpha$ denotes the angle between $UM$ and $MS$.

Fig. 1. A triangle UMS.

Hence

$$US^2 = (2x + 14)^2 + (14 + x)^2 - 2(2x + 14)(14 + x) \cos \alpha,$$ $$US^2 = (5x^2 - 4x^2 \cos \alpha) + (84x + 392)(1 - \cos \alpha),$$ $$US = \sqrt{(5x^2 - 4x^2 \cos \alpha) + (84x + 392)(1 - \cos \alpha)}.$$

In particular cases, when $\alpha = 0{}^\circ$ or $\alpha = 180{}^\circ$, the triangle UMS degenerates (see Fig. 2) and equation (2) can be simplified:

$$US = x, \text{ if } \alpha = 0{}^\circ;$$ $$US = 3x + 28, \text{ if } \alpha = 180{}^\circ.$$

Fig. 2. The cases of a degenerated triangle UMS.

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