To determine if the triangle with leg lengths 3 cm and 4 cm and a slope (hypotenuse) of 6 cm is a right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that for a right triangle with legs of length \(a\) and \(b\), and hypotenuse \(c\), the following relationship holds:
\[ a^2 + b^2 = c^2 \]
In this case:
- \(a = 3\) cm (one leg)
- \(b = 4\) cm (the other leg)
- \(c = 6\) cm (the hypotenuse)
Now we can calculate \(a^2 + b^2\) and \(c^2\):
\[ a^2 = 3^2 = 9 \]
\[ b^2 = 4^2 = 16 \]
Now, add these values together:
\[ a^2 + b^2 = 9 + 16 = 25 \]
Next, we calculate \(c^2\):
\[ c^2 = 6^2 = 36 \]
Now, we compare \(a^2 + b^2\) with \(c^2\):
\[ a^2 + b^2 = 25 \]
\[ c^2 = 36 \]
Since \(25 \neq 36\), the condition of the Pythagorean theorem is not satisfied, and therefore, the triangle with legs 3 cm and 4 cm and a hypotenuse of 6 cm is not a right triangle.
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