Understanding Triangle Types Based on Side Lengths

This question asks us to identify the type of triangle based on the relationship between the squares of its side lengths. We are given a triangle with sides x, y, and z, where z is the largest side. The specific condition provided is \(x^2 + y^2 > z^2\).

Classifying Triangles by Side Lengths

Triangles can be classified as acute, obtuse, or right-angled based on the relationship between the square of the longest side and the sum of the squares of the other two sides. Let's assume 'c' is the longest side of a triangle with sides 'a', 'b', and 'c'. The rules are based on a variation of the Pythagorean theorem:

• If \(a^2 + b^2 = c^2\), the triangle is a right-angled triangle. The angle opposite the longest side (c) is exactly 90 degrees.
• If \(a^2 + b^2 < c^2\), the triangle is an obtuse-angled triangle. The angle opposite the longest side (c) is greater than 90 degrees.
• If \(a^2 + b^2 > c^2\), the triangle is an acute-angled triangle. All three angles in the triangle are less than 90 degrees. This is true even if c is the longest side, as long as the sum of squares of the other two sides is greater than the square of the longest side.

Applying the Given Condition to Determine Triangle Type

In the given problem, the sides are x, y, and z, and z is specified as the largest side. The condition given is \(x^2 + y^2 > z^2\). This condition directly matches the rule for classifying acute-angled triangles where the sum of the squares of the two shorter sides is greater than the square of the longest side.

Let's compare the given condition \(x^2 + y^2 > z^2\) with the rules for triangle types:

Since the given condition is \(x^2 + y^2 > z^2\), where z is the largest side, the triangle must be an acute-angled triangle.

Analyzing the Options

Let's look at the provided options:

1. Isosceles right angled triangle: This is a specific type of right triangle (two sides equal) and requires \(x^2 + y^2 = z^2\). The given condition is different.
2. Acute angled triangle: This type fits the condition \(x^2 + y^2 > z^2\) when z is the largest side.
3. Obtuse angled triangle: This type requires \(x^2 + y^2 < z^2\) when z is the largest side. The given condition is different.
4. Right angled triangle: This type requires \(x^2 + y^2 = z^2\) when z is the largest side. The given condition is different.

Based on our analysis, the condition \(x^2 + y^2 > z^2\) for a triangle where z is the largest side corresponds to an acute-angled triangle.

Conclusion

Given that x, y, and z are the sides of a triangle, z is the largest side, and the condition \(x^2 + y^2 > z^2\) holds, the triangle is an acute-angled triangle.

Revision Table: Triangle Classification

Additional Information: Triangle Properties

Besides classification by angles (acute, right, obtuse), triangles can also be classified by their side lengths:

• Equilateral triangle: All three sides are equal in length. All angles are 60 degrees (which are acute angles). An equilateral triangle is always acute-angled.
• Isosceles triangle: At least two sides are equal in length. The angles opposite the equal sides are also equal. An isosceles triangle can be acute, right, or obtuse.
• Scalene triangle: All three sides have different lengths. All three angles also have different measures. A scalene triangle can be acute, right, or obtuse.

The condition \(x^2 + y^2 > z^2\) with z being the largest side focuses specifically on the angle opposite the longest side, which is why it determines the angle-based classification (acute, obtuse, or right).