Question

Classify LMN by its side lengths and by its angles

Answer 1

Step-by-step explanation:

Based on the given side lengths, LM=5cm, LN=5cm, and MN=4cm, we can classify the triangle LMN by its side lengths and angles.

By Side Lengths:
Since all three sides of the triangle have different lengths, we classify it as a scalene triangle. A scalene triangle is a triangle in which all three sides have different lengths.

By Angles:
To determine the classification based on angles, we can use the Law of Cosines to find one of the angles. Let's find angle LMN using the Law of Cosines:

c² = a² + b² - 2ab * cos(C)
where c is the side opposite angle C.

In this case, a=5cm, b=4cm, and c=5cm.
Substituting the values, we have:

5² = 5² + 4² - 2 * 5 * 4 * cos(C)
25 = 25 + 16 - 40 * cos(C)
25 = 41 - 40 * cos(C)
40 * cos(C) = 41 - 25
40 * cos(C) = 16
cos(C) = 16/40
cos(C) = 0.4

Now, we can use the inverse cosine (cos⁻¹) function to find angle C:

C = cos⁻¹(0.4)
C ≈ 66.42 degrees

Since we know the value of angle C, we can determine the remaining angles using the Triangle Angle Sum Property, which states that the sum of the angles in a triangle is always 180 degrees.

Angle LNM = Angle LMN = 180 - (66.42 + 90)
Angle LNM ≈ 23.58 degrees

Therefore, based on the angle measurements, triangle LMN can be classified as an acute scalene triangle since all three angles are less than 90 degrees and all three sides have different lengths.

Answer 2

Triangle LMN is an isosceles triangle (option B) due to equal side lengths LM and LN. It is also an acute triangle (option E) since all its angles—L, M, and N—are less than 90 degrees.

Triangle LMN can be classified both by its side lengths and angles. In terms of side lengths, since LM = LN, it is classified as an isosceles triangle (option B).

Regarding its angles, all three angles (L, M, and N) are acute, making it an acute triangle (option E).

An isosceles triangle has at least two equal sides, and in LMN, LM and LN are equal.

Additionally, an acute triangle has all three angles measuring less than 90 degrees, which holds true for LMN. Therefore, options B and E accurately describe the classification of triangle LMN based on its side lengths and angles, respectively.