Question

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Answer 1

This triangle is obtuse and isosceles

Explanation:

An obtuse triangle is a triangle in which one of the angles is greater than 90°. You can clearly see that angle NPO is greater than a right angle.

An isosceles triangle is a triangle in which two sides are equal and two angles are equal. Sides NP and PO are equal because they both measure 4m, which necessarily means that the angles opposite them are congruent (angles PNO and NOP).

Answer 2

ΔNOP is obtuse triangle and isosceles triangle.

Explanation:

An acute triangle has three angles that each measure less than 90°.

An obtuse triangle is a triangle with one angle greater than 90°.

A right triangle is a triangle with one 90° angle represented by the symbol ∟.

From inspection of the triangle, it appears that angle P is greater than 90°, which means it is an obtuse triangle.

This can be confirmed by using the cosine rule to calculate angle P.

`\boxed{\begin{minipage}{7.6 cm}\underline{Cosine Rule (for finding angles)} \\\\$\cos(C)=(a^2+b^2-c^2)/(2ab)$\\\\\\where:\\ \phantom{ww}$\bullet$ $C$ is the angle. \\ \phantom{ww}$\bullet$ $a$ and $b$ are the sides adjacent the angle. \\ \phantom{ww}$\bullet$ $c$ is the side opposite the angle.\\\end{minipage}}`

Therefore:

`\implies \cos(P)=(4^2+4^2-7^2)/(2(4)(4))`

`\implies \cos(P)=-(17)/(32)`

`\implies P=\cos^(-1)\left(-(17)/(32)\right)`

`\implies P=122.08995...^(\circ)`

As 122.1° > 90°, this proves that triangle NOP is obtuse.

An equilateral triangle has three sides of equal length.

A scalene triangle has three sides of differing lengths.

An isosceles triangle has two sides of equal length.

Therefore, as NP = OP ≠ ON, the triangle has two sides of equal length and so it an isosceles triangle.