How can we find the angles of a triangle given their measures in the extended ratio of 3:4:8?

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To find the angles of a triangle when the measures are in an extended ratio such as 3:4:8, you can follow these steps:

1. **Understand the Extended Ratio**: The ratio 3:4:8 means that we can represent the angles of the triangle as 3x, 4x, and 8x, where x is a common multiplier.

2. **Use the Triangle Sum Theorem**: The sum of the angles in a triangle always equals 180 degrees. Therefore, you can write the equation:

     3x + 4x + 8x = 180

3. **Simplify the Equation**: Combine like terms:

     15x = 180

4. **Solve for x**: Divide both sides of the equation by 15:

     x = 12

5. **Find Each Angle**: Now substitute x back into the expressions for each angle:

  •      First angle = 3x = 3(12) = 36 degrees
  •      Second angle = 4x = 4(12) = 48 degrees
  •      Third angle = 8x = 8(12) = 96 degrees

6. **Check Your Work**: Make sure the angles add up to 180 degrees:

     36 + 48 + 96 = 180, which is correct!

Thus, the measures of the angles in the triangle are:

  • First angle: 36 degrees
  • Second angle: 48 degrees
  • Third angle: 96 degrees

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