Answer: The given triangle LMN is an obtuse-angled triangle.
Step-by-step explanation: We are given to use Pythagorean identities to prove whether ΔLMN is a right, acute, or obtuse triangle.
From the figure, we note that
in ΔLMN, LM = 5 units, MN = 13 units and LN = 14 units.
We know that a triangle with sides a units, b units and c units (a > b, c) is said to be
(i) Right-angled triangle if [tex]b^2+c^2=a^2,[/tex]
(ii) Acute-angled triangle if [tex]b^2+c^2>a^2,[/tex]
(iii) Obtuse-angled triangle if [tex]b^2+c^2<a^2.[/tex]
For the given triangle LMN, we have
a = 14, b = 13 and c = 5.
So,
[tex]b^2+c^2=13^2+5^2=169+25=194,\\\\a^2=14^2=196.[/tex]
Therefore, [tex]b^2+c^2<a^2.[/tex]
Thus, the given triangle LMN is an obtuse-angled triangle.
