Math Problem Statement

What is the length of side MN in an isosceles triangle where two angles are 63 degrees and the two sides opposite these angles are 3x and x+8?

Solution

To solve this problem, we need to use the properties of an isosceles triangle and basic geometry.

Step-by-step explanation:

  1. Identify the triangle's structure:

    • LMN\triangle LMN is an isosceles triangle, meaning two of its sides are equal.
    • L=N=63\angle L = \angle N = 63^\circ, indicating that the triangle is isosceles with the base LNLN, and the two equal sides are LMLM and MNMN.
  2. Sum of angles in a triangle:

    • The sum of interior angles in any triangle is always 180180^\circ.
    • Let M\angle M be the third angle. Since L=63\angle L = 63^\circ and N=63\angle N = 63^\circ: M=1806363=54.\angle M = 180^\circ - 63^\circ - 63^\circ = 54^\circ.
  3. Side lengths:

    • The length of side LM=3xLM = 3x and the length of side MN=x+8MN = x + 8.
    • Since LMN\triangle LMN is isosceles, the two sides LMLM and MNMN must be equal. 3x=x+8.3x = x + 8.
  4. Solve for xx: 3xx=8,3x - x = 8, 2x=8,2x = 8, x=4.x = 4.

  5. Find the length of MNMN:

    • Since MN=x+8MN = x + 8, substitute x=4x = 4: MN=4+8=12.MN = 4 + 8 = 12.

Thus, the length of MNMN is 12 units.

Answer:

The correct option is B. 12.

Would you like a deeper explanation of any step? Here are some related questions you might find interesting:

  1. What is the importance of knowing the properties of isosceles triangles in geometry?
  2. Can you explain how to solve triangles that are not isosceles or equilateral?
  3. How can you apply angle properties in different types of triangles?
  4. How would this problem change if the angles were different?
  5. What are some real-world applications of isosceles triangles?

Tip: Always verify if triangles are isosceles or equilateral before solving for sides and angles, as their properties simplify the solution.

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Math Problem Analysis

Mathematical Concepts

Algebra
Isosceles Triangle Properties
Sum of Angles in a Triangle

Formulas

Sum of angles in a triangle = 180 degrees
Equal sides in an isosceles triangle

Theorems

Isosceles Triangle Theorem

Suitable Grade Level

Grades 7-9