Short Answer 
The problem focuses on an isosceles triangle MNK with equal sides MN and NK, and an angle at N of 110º. The two equal angles at M and K are calculated to be 70º each, and using trigonometry, the length of side MN is found to be approximately 12.16 after applying the sine function.
Step 1: Understand the Triangle Properties
We start by recognizing that triangle MNK is an isosceles triangle based on the conditions provided. In this triangle, the lengths of sides MN and NK are equal, meaning we have two sides of the same length. Also, we know that the angle m√¢¬a¬†N is 110¬∫, which helps us determine the measures of the other two angles in the triangle.
Step 2: Calculate the Angles
Since triangle MNK is isosceles, the angles at points M and K will be equal. We can calculate these angles using the angle sum property of triangles, which states that the sum of all angles in a triangle is 180º. Thus, angle measures for M and K are calculated as follows:
- Sum of angles = 180º
- Angle at N = 110º
- Angles at M and K = (180¬∫ Р110¬∫) / 2 = 70¬∫
Step 3: Use Trigonometry to Find MN
To find the length of side MN, we can drop a perpendicular line from point N to side MK. This creates a right triangle where one leg measures 5/2 (half of MK) and the angle opposite to MN is 70º. We apply the sine function, which relates the opposite side to the hypotenuse:
- sin(70º) = MN / (5/2)
- Solving for MN gives MN = (5/2) * sin(70º)
- This calculation results in MN being approximately equal to 12.16.