To prove that points $A(-3,4)$, $B(1,1)$, and $C(7,9)$ form the vertices of a right triangle, we can use the distance formula to calculate the lengths of sides $AB$, $BC$, and $CA$ and then check if they satisfy the Pythagorean theorem for a right triangle.

Step 1: Calculate the Distance Between Each Pair of Points

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

1. Distance $AB$

For points $A(-3,4)$ and $B(1,1)$: $AB = \sqrt{(1 - (-3))^2 + (1 - 4)^2} = \sqrt{(1 + 3)^2 + (1 - 4)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

2. Distance $BC$

For points $B(1,1)$ and $C(7,9)$: $BC = \sqrt{(7 - 1)^2 + (9 - 1)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$

3. Distance $CA$

For points $C(7,9)$ and $A(-3,4)$: $CA = \sqrt{(7 - (-3))^2 + (9 - 4)^2} = \sqrt{(7 + 3)^2 + (9 - 4)^2} = \sqrt{10^2 + 5^2} = \sqrt{100 + 25} = \sqrt{125} = 5\sqrt{5}$

Step 2: Check the Pythagorean Theorem

To confirm a right triangle, one of the side lengths squared should equal the sum of the squares of the other two sides.

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

• $AB^2 = 5^2 = 25$
• $BC^2 = 10^2 = 100$
• $CA^2 = (5\sqrt{5})^2 = 25 \times 5 = 125$

Now check: $AB^2 + BC^2 = 25 + 100 = 125 = CA^2$

Since $AB^2 + BC^2 = CA^2$, the points $A$, $B$, and $C$ satisfy the Pythagorean theorem, confirming they form a right triangle with the right angle at point $B$.

Conclusion

The points $A(-3,4)$, $B(1,1)$, and $C(7,9)$ are the vertices of a right triangle, with the right angle located at $B$.

Would you like further details or have any questions?

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Additional Questions to Explore

1. How do we confirm that a triangle is isosceles given three points?
2. How would we find the area of the triangle with vertices $A$, $B$, and $C$?
3. How would we determine the angle at each vertex of the triangle?
4. Can we verify the type of triangle (acute, obtuse, or right) using slopes instead of distances?
5. What are other methods to determine if a triangle is right-angled without the distance formula?

Tip:

The Pythagorean theorem can confirm a right angle in any triangle when comparing the squares of its side lengths.