Question

Karen says that there is only one triangle with sides of 3 inches, 4 inches, and 5 inches. Mika says that there can be many different triangles with those measures. Who is correct? Explain your reasoning.

Short answer

Karen is correct; only one triangle can have the given sides (3, 4, 5) as it uniquely forms a right triangle.

Step-by-step solution

Understand the Type of Triangle

Given the side lengths of 3 inches, 4 inches, and 5 inches, we need to verify if these form a specific type of triangle. These can form a right triangle as they satisfy the Pythagorean theorem, where \(3^2 + 4^2 = 5^2\).

Evaluate the Uniqueness of the Triangle

In geometry, for a set of three side lengths (3, 4, 5) that form a triangle, as is the case with a right triangle, the shape and size are determined uniquely. There can't be another triangle with different angles but the same side lengths. This is because, with the given side lengths, only one combination satisfies both the triangle inequality and forms a right triangle.

Consider Rotations or Reflections

Even if we consider rotations or reflections of the triangle, these do not create new or different triangles—they are the same triangle in different orientations. Therefore, the overall shape remains the same.

Conclusion Based on Properties

Given that a triangle with sides 3 inches, 4 inches, and 5 inches is uniquely a right triangle and cannot be altered in size or shape, Karen is correct.

Key concepts

Triangle Inequality

The triangle inequality theorem is fundamental in geometry to determine whether three side lengths can form a triangle. It states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This can be formulated as:

• For sides a, b, and c, the inequalities must satisfy:
  • \(a + b > c\)
  • \(b + c > a\)
  • \(a + c > b\)

Applying this to the given side lengths of 3, 4, and 5 inches, the conditions hold:

• \(3 + 4 > 5\) is true.
• \(4 + 5 > 3\) is true.
• \(3 + 5 > 4\) is true.

This confirms that these lengths can indeed form a triangle. The triangle inequality not only helps in evaluating the possibility of a triangle but also ensures that the sides enclose a space. This property is essential for deducing whether constructing such a triangle is geometrically feasible.

Pythagorean Theorem

The Pythagorean theorem is a cornerstone in geometry, especially relevant here since we are dealing with a right triangle. It relates the lengths of the sides of a right triangle, stating that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
For side lengths 3, 4, and 5, this theorem can be expressed as:

• \(3^2 + 4^2 = 5^2\)
• \(9 + 16 = 25\)

Thus, these calculations confirm that the side lengths of 3, 4, and 5 make a perfect set for a right triangle. Recognizing these sides as a Pythagorean triple allows us to identify it uniquely as a right triangle. The Pythagorean theorem not only verifies the right angle's existence but also supports the conclusion that the dimensions given will always result in the same triangular shape and size.

Unique Triangle

When given a particular set of side lengths, such as 3, 4, and 5, the concept of a unique triangle comes into play. A unique triangle means that there is only one possible shape and size for the triangle with those exact side lengths.
Because these sides satisfy both the triangle inequality and the Pythagorean theorem, they outline only one specific triangle type—a right triangle. This unique configuration ensures that no other triangle can exist with different angles but the same side lengths.

• This happens because the angles in the triangle are completely determined by the side lengths. There can be no variation in angle measures, leading to only one definitive triangular shape.
• Even if you rotate or reflect the triangle, its inherent properties remain unaltered. It still represents the same triangle in geometric terms.

So, despite Mika's belief, there are not multiple triangles with sides 3, 4, and 5. Karen is correct in that the triangle is unique by shape and size due to these fixed side lengths.