Classifying a Triangle with Sides 6, 11, and 14: Acute, Obtuse, or Right Angle?

Triangles are often classified based on their angles or side lengths. One method to determine if a given triangle with side lengths of 6, 11, and 14 is acute, obtuse, or right is by comparing the squares of the side lengths. This article will walk you through the process and provide a detailed explanation.

Identifying the Sides and Their Squares

For a triangle with sides of lengths 6, 11, and 14, let's denote them as follows:

a 6 b 11 c 14

Since c is the longest side, it will play a pivotal role in our classification.

Calculate the Squares of the Sides

Now, calculate the squares of the lengths:

a^2 6^2 36 b^2 11^2 121 c^2 14^2 196

Compare the Sums and Determine the Type of Triangle

To determine if the triangle is acute, obtuse, or right, we compare the sum of the squares of the two shorter sides with the square of the longest side:

a^2 b^2 36 121 157 c^2 196

Compare a^2 b^2 with c^2:

157 196

Based on the comparison, we can determine the type of triangle:

If a^2 b^2 c^2, the triangle is obtuse. If a^2 b^2 c^2, the triangle is right. If a^2 b^2 c^2, the triangle is acute.

Since 157 196, the triangle is obtuse.

Properties of a Scalene Triangle

A triangle with sides of lengths 6, 11, and 14 is a scalene triangle. Let's examine the properties of this scalene triangle:

All angles of a scalene triangle are unequal. A scalene triangle has no line of symmetry. The angle opposite the longest side (14) will be the greatest angle, and the angle opposite the shortest side (6) will be the smallest angle. The given triangle cannot be divided into two identical halves; there is no line of symmetry.

Angles in a Scalene Triangle

In the triangle with sides 6, 11, and 14:

The greatest angle (let's call it E) is opposite the side of length 14 (let's call it GO). The smallest angle (let's call it O) is opposite the side of length 6 (let's call it GE).

A scalene triangle can be acute-angled, obtuse-angled, or right-angled. In this case, our triangle has been determined to be obtuse-angled.

Conclusion

The triangle with sides 6, 11, and 14 is an obtuse triangle. This is confirmed by the fact that the sum of the squares of the two shorter sides (6 and 11) is less than the square of the longest side (14). This aligns with the property that in an obtuse triangle, the square of the longest side is greater than the sum of the squares of the other two sides.