Key Concepts and Formulas

• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
• Pythagorean Theorem: In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides: $a^2 + b^2 = c^2$.
• Area of a Triangle (given coordinates): Area = $\frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$.
• Area of a Right-Angled Triangle: Area = $\frac{1}{2} \times \text{base} \times \text{height}$.

Step-by-Step Solution

Step 1: Identify the Right Angle and Apply the Pythagorean Theorem

Given vertices $A(h,k)$, $B(1,1)$, and $C(2,1)$, with $AC$ as the hypotenuse. This means the right angle is at vertex $B$. Therefore, by the Pythagorean Theorem: $AB^2 + BC^2 = AC^2$ This equation will relate the unknown coordinates $h$ and $k$.

Step 2: Calculate the Squares of the Lengths of Each Side

Use the distance formula to find the squared lengths:

• Length of $AB$: $AB^2 = (h-1)^2 + (k-1)^2$
• Length of $BC$: $BC^2 = (2-1)^2 + (1-1)^2 = 1^2 + 0^2 = 1$ Explanation: $BC$ is a horizontal line segment since $B$ and $C$ have the same $y$-coordinate.
• Length of $AC$: $AC^2 = (h-2)^2 + (k-1)^2$

Step 3: Substitute Lengths into the Pythagorean Theorem to Find 'h'

Substitute the expressions for $AB^2$, $BC^2$, and $AC^2$ into the Pythagorean equation: $(h-1)^2 + (k-1)^2 + 1 = (h-2)^2 + (k-1)^2$ This substitution forms an equation that relates $h$ and $k$.

Subtract $(k-1)^2$ from both sides: $(h-1)^2 + 1 = (h-2)^2$ Expand the squared terms: $h^2 - 2h + 1 + 1 = h^2 - 4h + 4$ $h^2 - 2h + 2 = h^2 - 4h + 4$ Subtract $h^2$ from both sides: $-2h + 2 = -4h + 4$ Rearrange the terms to solve for $h$: $2h = 2$ $h = 1$ Explanation: This step determines the value of $h$, which is essential for calculating the area later.

Step 4: Use the Area of the Triangle to Find 'k'

Since we know $A = (1, k)$, $B = (1, 1)$, and $C = (2, 1)$, we can use the area formula with coordinates: Area $= \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$ Substituting the coordinates of $A$, $B$, and $C$: $1 = \frac{1}{2} |1(1 - 1) + 1(1 - k) + 2(k - 1)|$ $1 = \frac{1}{2} |0 + 1 - k + 2k - 2|$ $1 = \frac{1}{2} |k - 1|$ Multiply both sides by 2: $2 = |k - 1|$ Explanation: We now have an absolute value equation to solve for $k$.

Step 5: Solve for k

The absolute value equation $|k - 1| = 2$ gives us two possible cases:

Case 1: $k - 1 = 2$ $k = 3$

Case 2: $k - 1 = -2$ $k = -1$

Therefore, the possible values for $k$ are $3$ and $-1$.

Common Mistakes & Tips

• Remember to use the correct area formula. Using the determinant form or $\frac{1}{2}bh$ carefully, ensuring the base and height are perpendicular.
• When taking the square root, remember that the distance is always positive. It can be helpful to use absolute value notation.
• Be careful with algebraic manipulations, especially when expanding and simplifying equations.

Summary

We used the Pythagorean theorem and the area of a triangle formula to find the possible values of $k$. By identifying the right angle and using the given area, we set up equations that allowed us to solve for $h$ and then $k$. The possible values for $k$ are $-1$ and $3$.

Final Answer

The final answer is $\boxed{\{-1, 3\}}$, which corresponds to option (A).