Identifying Obtuse-Angled Triangles from Side Lengths

To determine if a set of three lengths can form an obtuse-angled triangle, we need to check two conditions:

1. Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If this condition is not met, the lengths cannot form a triangle at all.
2. Classification based on side lengths: If the lengths $a$, $b$, and $c$ form a triangle, where $c$ is the longest side, the type of triangle is determined by comparing $a^2 + b^2$ with $c^2$:
   • If $a^2 + b^2 = c^2$, it is a right-angled triangle.
   • If $a^2 + b^2 > c^2$, it is an acute-angled triangle.
   • If $a^2 + b^2 < c^2$, it is an obtuse-angled triangle.

We will check each set of lengths provided in the options to see which one forms an obtuse-angled triangle.

Analyzing Option 1: Sides 15 cm, 62 cm, 64 cm

Let the sides be $a=15$, $b=62$, and $c=64$. The longest side is $c=64$.

• Check Triangle Inequality:
  • $15 + 62 = 77$. Is $77 > 64$? Yes.
  • $15 + 64 = 79$. Is $79 > 62$? Yes.
  • $62 + 64 = 126$. Is $126 > 15$? Yes.
  Since all conditions are met, these lengths can form a triangle.
• Classify the Triangle: We compare $a^2 + b^2$ with $c^2$.
  • $a^2 = 15^2 = 225$
  • $b^2 = 62^2 = 3844$
  • $a^2 + b^2 = 225 + 3844 = 4069$
  • $c^2 = 64^2 = 4096$
  Comparing $4069$ and $4096$: $4069 < 4096$. Since $a^2 + b^2 < c^2$, this is an obtuse-angled triangle.

Analyzing Option 2: Sides 17 cm, 64 cm, 66 cm

Let the sides be $a=17$, $b=64$, and $c=66$. The longest side is $c=66$.

• Check Triangle Inequality: $17 + 64 = 81$. Is $81 > 66$? Yes. The other inequalities will also hold as the third side is smaller. These lengths can form a triangle.
• Classify the Triangle: We compare $a^2 + b^2$ with $c^2$.
  • $a^2 = 17^2 = 289$
  • $b^2 = 64^2 = 4096$
  • $a^2 + b^2 = 289 + 4096 = 4385$
  • $c^2 = 66^2 = 4356$
  Comparing $4385$ and $4356$: $4385 > 4356$. Since $a^2 + b^2 > c^2$, this is an acute-angled triangle.

Analyzing Option 3: Sides 16 cm, 63 cm, 65 cm

Let the sides be $a=16$, $b=63$, and $c=65$. The longest side is $c=65$.

• Check Triangle Inequality: $16 + 63 = 79$. Is $79 > 65$? Yes. These lengths can form a triangle.
• Classify the Triangle: We compare $a^2 + b^2$ with $c^2$.
  • $a^2 = 16^2 = 256$
  • $b^2 = 63^2 = 3969$
  • $a^2 + b^2 = 256 + 3969 = 4225$
  • $c^2 = 65^2 = 4225$
  Comparing $4225$ and $4225$: $4225 = 4225$. Since $a^2 + b^2 = c^2$, this is a right-angled triangle (it's a Pythagorean triple).

Analyzing Option 4: Sides 18 cm, 65 cm, 67 cm

Let the sides be $a=18$, $b=65$, and $c=67$. The longest side is $c=67$.

• Check Triangle Inequality: $18 + 65 = 83$. Is $83 > 67$? Yes. These lengths can form a triangle.
• Classify the Triangle: We compare $a^2 + b^2$ with $c^2$.
  • $a^2 = 18^2 = 324$
  • $b^2 = 65^2 = 4225$
  • $a^2 + b^2 = 324 + 4225 = 4549$
  • $c^2 = 67^2 = 4489$
  Comparing $4549$ and $4489$: $4549 > 4489$. Since $a^2 + b^2 > c^2$, this is an acute-angled triangle.

Based on the analysis, only the set of lengths 15 cm, 62 cm, and 64 cm forms an obtuse-angled triangle.

Revision Table: Triangle Classification

Additional Information: Understanding Triangle Types

Triangles are fundamental shapes in geometry. They can be classified based on their side lengths (e.g., equilateral, isosceles, scalene) or based on their angles (acute, right, obtuse).

• An acute-angled triangle has all three angles less than 90 degrees.
• A right-angled triangle has exactly one angle equal to 90 degrees. The side opposite the right angle is called the hypotenuse, and it is always the longest side.
• An obtuse-angled triangle has exactly one angle greater than 90 degrees. The side opposite the obtuse angle is the longest side.

The relationship between the squares of the side lengths ($a^2 + b^2$ vs $c^2$) is a direct extension of the Pythagorean theorem, which specifically applies to right-angled triangles. This extended relationship allows us to determine the type of triangle even when it's not a right triangle, simply by knowing its side lengths.