To determine if the triangle with vertices $X(4, -3)$, $Y(9, 1)$, and $Z(0, 2)$ is a right triangle, we can use two different methods:

1. Using the Distance Formula and Pythagorean Theorem:
2. Using the Slope Method to check for Perpendicularity:

Method 1: Using the Distance Formula and Pythagorean Theorem

First, we calculate the lengths of all three sides using the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step 1: Calculate $XY$

$X(4, -3) \quad Y(9, 1)$ $XY = \sqrt{(9 - 4)^2 + (1 - (-3))^2}$ $XY = \sqrt{5^2 + 4^2}$ $XY = \sqrt{25 + 16}$ $XY = \sqrt{41}$

Step 2: Calculate $YZ$

$Y(9, 1) \quad Z(0, 2)$ $YZ = \sqrt{(0 - 9)^2 + (2 - 1)^2}$ $YZ = \sqrt{(-9)^2 + 1^2}$ $YZ = \sqrt{81 + 1}$ $YZ = \sqrt{82}$

Step 3: Calculate $ZX$

$Z(0, 2) \quad X(4, -3)$ $ZX = \sqrt{(4 - 0)^2 + (-3 - 2)^2}$ $ZX = \sqrt{4^2 + (-5)^2}$ $ZX = \sqrt{16 + 25}$ $ZX = \sqrt{41}$

Next, we check if the Pythagorean theorem holds: $a^2 + b^2 = c^2$

Let's assume $XY$ and $ZX$ are the legs, and $YZ$ is the hypotenuse: $(\sqrt{41})^2 + (\sqrt{41})^2 = (\sqrt{82})^2$ $41 + 41 = 82$ $82 = 82$

Since the Pythagorean theorem holds, the triangle $XYZ$ is a right triangle.

Method 2: Using the Slope Method to Check for Perpendicularity

We can find the slopes of the lines $XY$, $YZ$, and $ZX$ and check if any two are negative reciprocals (which indicates perpendicularity).

Step 1: Calculate the slope of $XY$

$X(4, -3) \quad Y(9, 1)$ $m_{XY} = \frac{1 - (-3)}{9 - 4}$ $m_{XY} = \frac{4}{5}$

Step 2: Calculate the slope of $YZ$

$Y(9, 1) \quad Z(0, 2)$ $m_{YZ} = \frac{2 - 1}{0 - 9}$ $m_{YZ} = \frac{1}{-9}$ $m_{YZ} = -\frac{1}{9}$

Step 3: Calculate the slope of $ZX$

$Z(0, 2) \quad X(4, -3)$ $m_{ZX} = \frac{-3 - 2}{4 - 0}$ $m_{ZX} = \frac{-5}{4}$

To check for perpendicularity:

• $m_{XY} \cdot m_{YZ} = \frac{4}{5} \cdot -\frac{1}{9} = -\frac{4}{45}$ (not equal to -1)
• $m_{YZ} \cdot m_{ZX} = -\frac{1}{9} \cdot -\frac{5}{4} = \frac{5}{36}$ (not equal to -1)
• $m_{ZX} \cdot m_{XY} = -\frac{5}{4} \cdot \frac{4}{5} = -1$ (equal to -1)

Since the product of the slopes of $ZX$ and $XY$ is $-1$, they are perpendicular, confirming that $XYZ$ is a right triangle.

Conclusion

Using both the Pythagorean theorem and the slope method, we have confirmed that the triangle with vertices $X(4, -3)$, $Y(9, 1)$, and $Z(0, 2)$ is indeed a right triangle.

Would you like any further details or have any questions about the solution?

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Here are 8 related questions for further practice:

1. How do you find the area of a right triangle given its side lengths?
2. How can you determine if a triangle is isosceles using its vertices?
3. What is the equation of the line passing through $X$ and $Y$?
4. How do you find the circumcenter of a triangle given its vertices?
5. What is the equation of the circle circumscribed around triangle $XYZ$?
6. How do you find the orthocenter of a triangle?
7. Can you determine if three points form an equilateral triangle? How?
8. How do you find the centroid of a triangle given its vertices?

Tip: When checking for right angles using slopes, remember that the slopes of perpendicular lines multiply to $-1$.