91Ó°ÊÓ

The vertices of a triangle are essentially the three corner points that define the shape and size of the triangle. Each vertex is represented as a coordinate point in a Cartesian plane, as seen in the problem statement where the vertices are given as \( A(h, k) \), \( B(1, 1) \), and \( C(2, 1) \). These points are used to determine various properties of the triangle, such as the length of sides, whether the triangle is right-angled, and the area of the triangle.
Understanding the coordinates of these vertices helps in developing a strategy to solve problems like determining the possible values of a variable in the vertex coordinates. It also aids in the application of geometric formulas, such as the area formula, to determine specific characteristics.

• Coordinate Representation: Each vertex is represented with an \((x, y)\) coordinate, denoting its horizontal and vertical position on the plane.
• Positioning: The exercise specifies that \( B \) and \( C \) are vertical to each other on the \( x \)-axis, hinting they are part of the base of the right-angle triangle.

In the context of a right-angled triangle, the hypotenuse is the longest side, opposite the right angle. It holds significant importance because it can be calculated using the Pythagorean Theorem: \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.
In this problem, the hypotenuse is the line segment \( AC \), connecting the points \( A(h, k) \) and \( C(2, 1) \). As the triangle is identified as right-angled, determining the relationship between these points revolves around understanding how the hypotenuse fits with the problem constraints, such as area.

• Identifying the Hypotenuse: Always opposite the right angle in a triangle.
• Pythagorean Theorem Utility: Helpful in solving relationships between triangle sides, but not directly needed for solving this particular exercise about area and vertex position.

The area of a triangle can be calculated using the vertex coordinates through the formula:\[\frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| = Area\]This method involves managing the algebraic relationships between vertex coordinates. Here, the problem specifies the area as \( 1 \), and we can use the known vertices \( B(1, 1) \) and \( C(2, 1) \) to substitute into the formula and solve for the unknown \( k \).
This is how we arrive at the simplified absolute value equation that eventually leads to finding the potential values of \( k \). Thorough understanding here involves practicing how to plug-in values and simplify the formula to extract practical solutions.

• Area Formula: Crucial for problems involving geometrical constraints, especially when specific coordinates are variable.
• Substitution Process: Carefully replace known coordinates into the formula to mirror the actual geometry and solve for unknowns.