To prove that a triangle is a right triangle, we can use the distance formula to compute the lengths of its sides and then check if the Pythagorean theorem holds for those lengths. The Pythagorean theorem states that for a right triangle, the square of the length of the hypotenuse (longest side) equals the sum of the squares of the lengths of the other two sides.

The vertices of the triangle are $A(6,1)$, $B(10,9)$, and $C(-6,7)$.

Step 1: Calculate the distances between each pair of points.

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Distance $AB$:

$AB = \sqrt{(10 - 6)^2 + (9 - 1)^2} = \sqrt{(4)^2 + (8)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}$

Distance $BC$:

$BC = \sqrt{(-6 - 10)^2 + (7 - 9)^2} = \sqrt{(-16)^2 + (-2)^2} = \sqrt{256 + 4} = \sqrt{260} = 2\sqrt{65}$

Distance $CA$:

$CA = \sqrt{(6 - (-6))^2 + (1 - 7)^2} = \sqrt{(6 + 6)^2 + (-6)^2} = \sqrt{12^2 + (-6)^2} = \sqrt{144 + 36} = \sqrt{180} = 6\sqrt{5}$

Step 2: Check the Pythagorean theorem.

The Pythagorean theorem is satisfied if the sum of the squares of the two shorter sides equals the square of the longest side. Let's check:

• The sides are $AB = 4\sqrt{5}$, $BC = 2\sqrt{65}$, and $CA = 6\sqrt{5}$. Clearly, $BC = 2\sqrt{65}$ is the longest side, so it should be the hypotenuse.

Let's check if: $AB^2 + CA^2 = BC^2$

Calculate $AB^2 + CA^2$:

$AB^2 = (4\sqrt{5})^2 = 16 \times 5 = 80$ $CA^2 = (6\sqrt{5})^2 = 36 \times 5 = 180$ $AB^2 + CA^2 = 80 + 180 = 260$

Calculate $BC^2$:

$BC^2 = (2\sqrt{65})^2 = 4 \times 65 = 260$

Since $AB^2 + CA^2 = BC^2$, the Pythagorean theorem holds, and the triangle is a right triangle.

Would you like more details or further questions? Here are 5 related questions for practice:

1. How can you classify triangles based on their side lengths?
2. What are the coordinates of the centroid of this triangle?
3. How would the method change if you wanted to check if the triangle is equilateral?
4. Can you derive the distance formula from the Pythagorean theorem?
5. How do you find the equation of the line passing through two points in a plane?

Tip: The Pythagorean theorem applies to all right triangles, but it can also help you identify if a triangle is acute or obtuse by comparing the sum of the squares of the shorter sides.