To prove that $(4, 4)$, $(3, 5)$, and $(-1, -1)$ are the vertices of a right triangle without directly using the Pythagorean Theorem, we will rely on the concept of slopes of lines. If two sides of the triangle are perpendicular, their slopes will multiply to $-1$.

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Steps to Prove:

1. Calculate the slopes of the lines formed by the vertices:

   • Slope between $(4, 4)$ and $(3, 5)$:
     $m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 4}{3 - 4} = \frac{1}{-1} = -1$

   • Slope between $(3, 5)$ and $(-1, -1)$:
     $m_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 5}{-1 - 3} = \frac{-6}{-4} = \frac{3}{2}$

   • Slope between $(-1, -1)$ and $(4, 4)$:
     $m_3 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-1)}{4 - (-1)} = \frac{5}{5} = 1$

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2. Check if any two slopes are perpendicular: Two lines are perpendicular if the product of their slopes is $-1$.

   • $m_1 \cdot m_2 = (-1) \cdot \frac{3}{2} = -\frac{3}{2}$ → Not perpendicular.
   • $m_2 \cdot m_3 = \frac{3}{2} \cdot 1 = \frac{3}{2}$ → Not perpendicular.
   • $m_1 \cdot m_3 = (-1) \cdot 1 = -1$ → Perpendicular!

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3. Conclusion: Since the slopes of the lines between $(4, 4) \to (3, 5)$ and $(-1, -1) \to (4, 4)$ multiply to $-1$, these two sides are perpendicular. Hence, the triangle formed by the vertices is a right triangle.

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Would you like additional clarification or further exploration into this problem?

Related Questions:

1. How do you calculate the area of this triangle using vertices?
2. What is the length of each side of the triangle using the distance formula?
3. How do you classify other triangles by their slopes or angles?
4. Can we prove the same result using vector dot products?
5. How would you graphically verify that this triangle is a right triangle?

Tip:

Using slopes is an efficient way to check perpendicularity and prove the presence of a right angle geometrically.