# The sides of triangle have lengths 7,5.1, and 6.2. What

The sides of triangle have lengths 7,5.1, and 6.2. What kind of triangle is it?
Find f.

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Roger Noah
a) Given sides of triangle have lengths 7,5.1, and 6.2
Since all the sides are different.

Let $$\displaystyle{c}={7}$$ (largest side)
$$\displaystyle{a}={5.1}$$ and $$\displaystyle{b}={6.2}$$
$$\displaystyle{c}^{{2}}={7}^{{2}}={49}{<}{\left({5.1}\right)}^{{2}}+{\left({6.2}\right)}^{{2}}={a}^{{2}}+{b}^{{2}}$$</span>
$$\displaystyle{c}^{{2}}{<}{a}^{{2}}+{b}^{{2}}$$</span>
if $$\displaystyle{c}^{{2}}{<}{a}^{{2}}+{b}^{{2}}$$</span>, then the triangle is an acute triangle
Acute b) Since $$\displaystyle{m}\angle{A}{C}{B}={m}\angle{A}{B}{C}={4}{s}$$

$$\displaystyle\Rightarrow{A}{B}={A}{C}={f}$$
$$\displaystyle{A}{B}={f}$$
$$\displaystyle{9}{n}\triangle{A}{B}{C},{A}{C}^{{2}}={A}{B}^{{2}}+{A}{C}^{{2}}$$
$$\displaystyle={f}^{{2}}+{f}^{{2}}={2}{f}^{{2}}$$
$$\displaystyle\Rightarrow{4}={2}{f}^{{2}}\Rightarrow{f}^{{2}}={2}$$
$$\displaystyle{f}=\sqrt{{{2}}}={1.414}$$
$$\displaystyle{f}={1.414}$$