- Verify the triangle inequality: \(5 + 12 > 13\).
- Calculate the squares of the sides: \(5^2 = 25\), \(12^2 = 144\), and \(13^2 = 169\).
- Check the Pythagorean theorem: \(25 + 144 = 169\).
- Conclude that the triangle is a right triangle because \(5^2 + 12^2 = 13^2\).
- Final answer: \(\boxed{\text{The triangle is a right triangle because } 5^2 + 12^2 = 13^2.}\)
Explanation
- Analysis of side lengths and existence of the triangle
To determine if a triangle with side lengths \(5 \text{ cm}\), \(12 \text{ cm}\), and \(13 \text{ cm}\) is a right triangle, we analyze the relationship between the squares of its side lengths using the Pythagorean Theorem. A triangle is a right triangle if and only if the sum of the squares of the two shorter sides is equal to the square of the longest side.
Given sides are \(a = 5\), \(b = 12\), and \(c = 13\). The longest side is \(c = 13\). We first ensure these lengths form a valid triangle by checking the triangle inequality, which states that the sum of the two shorter sides must be greater than the longest side:
\[5 + 12 = 17 > 13\]
Since the condition is satisfied, a triangle with these side lengths exists.
- Verification using the Pythagorean theorem
Next, we calculate the square of each side length:
\[5^2 = 25\]
\[12^2 = 144\]
\[13^2 = 169\]
Now, calculate the sum of the squares of the two shorter sides:
\[5^2 + 12^2 = 25 + 144 = 169\]
Comparing this sum to the square of the longest side:
\[169 = 169\]
This demonstrates that the Pythagorean relationship \(a^2 + b^2 = c^2\) holds true for this triangle.
- Conclusion
Since the relationship \(5^2 + 12^2 = 13^2\) is satisfied, the triangle is indeed a right triangle. The correct explanation is: "The triangle is a right triangle because \(5^2 + 12^2 = 13^2\)."
Examples
Understanding whether a triangle is a right triangle is crucial in various fields like carpentry and construction. For instance, when framing a corner for a wall, a carpenter can verify that the corner is perfectly square (90 degrees) by measuring a triangle with sides in a 3:4:5 ratio or any multiple of it (like 5, 12, 13). By ensuring the side lengths satisfy the equation \(a^2 + b^2 = c^2\), they can guarantee a precise, right-angled structure, preventing issues with uneven floors or walls later in the building process.