The angles of a triangle are in a ratio what is the measurement of the 2nd largest angle?

Mathematics
1

the angles of a triangle are in a ratio what is the measurement of the 2nd largest angle?

2

The angles of a triangle are in a ratio. What is the measurement of the 2nd largest angle?

Answer:
To find the measurement of the 2nd largest angle in a triangle where the angles are in a given ratio, we follow a systematic approach. We will explore a general method applicable to any ratio provided.

Step-by-Step Solution:

  1. Define the Angles in Terms of the Ratio:

    • Let’s denote the angles of the triangle as (A), (B), and (C) which are in the ratio (m:n:p).
    • This implies the angles can be written as:
      A = m \cdot k, \quad B = n \cdot k, \quad C = p \cdot k
      where (k) is a common multiplier.

  2. Sum of Angles in a Triangle:

    • We know the sum of the angles in any triangle is (180^\circ).
      A + B + C = 180^\circ
    • Substituting the expressions for (A), (B), and (C):
      m \cdot k + n \cdot k + p \cdot k = 180^\circ
    • Factor out (k):
      k(m + n + p) = 180^\circ

  3. Solve for (k):

    • Rearrange to solve for the common multiplier (k):
      k = \frac{180^\circ}{m + n + p}

  4. Calculate Each Angle:

    • Using the value of (k), we can now find the measures of each angle:

      A = m \cdot k = m \left(\frac{180^\circ}{m + n + p}\right)
      B = n \cdot k = n \left(\frac{180^\circ}{m + n + p}\right)
      C = p \cdot k = p \left(\frac{180^\circ}{m + n + p}\right)

  5. Determine the 2nd Largest Angle:

    • Arrange the angles (A), (B), and (C) in descending order to determine the 2nd largest, considering the specific values of (m), (n), and (p).

Example:

Assume the angles are in the ratio (2:3:5).

  1. Express the Angles:

    • Let:
      A = 2k, \quad B = 3k, \quad C = 5k

  2. Sum of Angles:

    • Set up the equation:
      2k + 3k + 5k = 180^\circ
      10k = 180^\circ
      k = 18^\circ

  3. Calculate Individual Angles:

    • Calculate each angle:

      A = 2 \cdot 18^\circ = 36^\circ
      B = 3 \cdot 18^\circ = 54^\circ
      C = 5 \cdot 18^\circ = 90^\circ

  4. Identify the 2nd Largest Angle:

    • Arrange the angles:
      \{90^\circ, 54^\circ, 36^\circ\}
    • The 2nd largest angle is (54^\circ).

Final Answer:
The measurement of the 2nd largest angle is (\boxed{54^\circ}).

Conceptual Understanding:

This process demonstrates the importance of understanding proportions and the invariant sum of triangle angles. By expressing each angle through a common multiplier and leveraging the constraint that angles add up to (180^\circ), we solve for the individual measures and identify the required angle.

Recap:

  • Define angles based on given ratio.
  • Use triangle sum property.
  • Solve for common multiplier.
  • Calculate each angle.
  • Determine the 2nd largest angle.

By following these steps, one can confidently determine specific angle measures in any proportional triangle scenario.