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 \(\displaystyle{m}\angle{A}{C}{B}={m}\angle{A}{B}{C}={4}{s}\)

\(\displaystyle\Rightarrow{A}{B}={A}{C}={f}\)

\(\displaystyle{A}{B}={f}\)

\(\displaystyle{9}{n}\triangle{A}{B}{C},{A}{C}^{{2}}={A}{B}^{{2}}+{A}{C}^{{2}}\)

\(\displaystyle={f}^{{2}}+{f}^{{2}}={2}{f}^{{2}}\)

\(\displaystyle\Rightarrow{4}={2}{f}^{{2}}\Rightarrow{f}^{{2}}={2}\)

\(\displaystyle{f}=\sqrt{{{2}}}={1.414}\)

\(\displaystyle{f}={1.414}\)

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 \(\displaystyle{m}\angle{A}{C}{B}={m}\angle{A}{B}{C}={4}{s}\)

\(\displaystyle\Rightarrow{A}{B}={A}{C}={f}\)

\(\displaystyle{A}{B}={f}\)

\(\displaystyle{9}{n}\triangle{A}{B}{C},{A}{C}^{{2}}={A}{B}^{{2}}+{A}{C}^{{2}}\)

\(\displaystyle={f}^{{2}}+{f}^{{2}}={2}{f}^{{2}}\)

\(\displaystyle\Rightarrow{4}={2}{f}^{{2}}\Rightarrow{f}^{{2}}={2}\)

\(\displaystyle{f}=\sqrt{{{2}}}={1.414}\)

\(\displaystyle{f}={1.414}\)