Question

1. name the triangle with the following characteristics. sides: 2 cm, 3 cm, 4 cm; angle: 29°.

right isosceles triangle
acute scalene triangle
obtuse scalene triangle
right scalene triangle

Explanation:

Step1: Analyze side lengths

The side lengths are 2 cm, 3 cm, 4 cm. All sides are of different lengths, so the triangle is scalene (since scalene triangles have all sides unequal).

Step2: Analyze the angle

One angle is \(29^\circ\). Let's check the largest angle using the Law of Cosines. For a triangle with sides \(a = 2\), \(b = 3\), \(c = 4\) (where \(c\) is the longest side), the Law of Cosines states \(c^{2}=a^{2}+b^{2}-2ab\cos C\). Rearranging for \(\cos C\): \(\cos C=\frac{a^{2}+b^{2}-c^{2}}{2ab}\). Substituting the values: \(\cos C=\frac{2^{2}+3^{2}-4^{2}}{2\times2\times3}=\frac{4 + 9-16}{12}=\frac{-3}{12}=-\frac{1}{4}\approx - 0.25\). Then \(C=\arccos(- 0.25)\approx104.48^\circ\), which is greater than \(90^\circ\), so the triangle is obtuse.

Step3: Combine the classifications

Since the triangle is scalene (all sides different) and obtuse (one angle greater than \(90^\circ\)), it is an obtuse scalene triangle.

Answer:

obtuse scalene triangle