what will be the maximum obtuse angle made in isoceles triangle when perimeter “p” is given

To determine the maximum obtuse angle in an isosceles triangle given a perimeter "p," we need to delve into some geometric principles. An isosceles triangle has two sides of equal length, and the angles opposite those sides are also equal. The third angle, which is the vertex angle, can be obtuse, right, or acute depending on the lengths of the sides.

Understanding the Triangle's Properties

In an isosceles triangle, let’s denote the lengths of the two equal sides as "a" and the base as "b." The perimeter "p" can be expressed as:

• p = 2a + b

From this equation, we can express "b" in terms of "a" and "p":

• b = p - 2a

Finding the Maximum Obtuse Angle

For an angle to be obtuse, it must be greater than 90 degrees. In an isosceles triangle, the obtuse angle is typically the vertex angle, which is opposite the base "b." To maximize this angle, we need to minimize the base "b" while ensuring that the triangle inequality holds true.

Applying the Triangle Inequality

The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side. For our isosceles triangle, this gives us the following conditions:

• 2a > b
• a + b > a (which simplifies to b > 0)
• a + a > b (which is the same as the first condition)

Substituting for "b," we get:

• 2a > p - 2a

This simplifies to:

• 4a > p
• a > p/4

Calculating the Angle

To find the angle, we can use the cosine rule, which states:

• c² = a² + b² - 2ab * cos(C)

In our case, we want to find the angle C (the vertex angle) when "b" is minimized. As "b" approaches zero, the triangle becomes increasingly narrow, and the angle C approaches 180 degrees. However, we need to ensure that the triangle remains valid, so we must keep "b" positive.

Maximizing the Obtuse Angle

To maximize the obtuse angle, we can set "b" to a very small positive value. For practical purposes, let’s consider "b" approaching zero. In this scenario, the angle C approaches 120 degrees, which is the maximum obtuse angle possible in an isosceles triangle given a fixed perimeter.

Conclusion

In summary, the maximum obtuse angle that can be formed in an isosceles triangle with a given perimeter "p" is 120 degrees. This occurs when the base "b" is minimized, while still satisfying the triangle inequality. Understanding these relationships helps in visualizing how the properties of triangles interact with one another.