The area of a triangle is 5 . Two of its vertices are (2,1)...

The area of a triangle is 5 . Two of its vertices are $(2,1)$ and $(3,-2)$, the third vertex satisfies $y=x+3$. The coordinates of the third vertex can be (a) $\left(-\frac{3}{2}, \frac{3}{2}\right)$ (b) $\left(\frac{3}{2},-\frac{3}{2}\right)$ (c) $\left(\frac{7}{2}, \frac{13}{2}\right)$ (d) $\left(-\frac{1}{4}, \frac{11}{4}\right)$.

The area of a triangle is 5 . Two of its vertices are $(2,1)$ and $(3,-2)$, the third vertex satisfies $y=x+3$. The coordinates of the third vertex can be
(a) $\left(-\frac{3}{2}, \frac{3}{2}\right)$
(b) $\left(\frac{3}{2},-\frac{3}{2}\right)$
(c) $\left(\frac{7}{2}, \frac{13}{2}\right)$
(d) $\left(-\frac{1}{4}, \frac{11}{4}\right)$.

Key Concepts

Triangle Area Using the Determinant Formula

This concept involves calculating the area of a triangle using its vertex coordinates via the determinant formula. The formula is based on the absolute value of a determinant of a matrix constructed from the coordinates of the vertices, and it provides a straightforward method to calculate the area using algebraic methods. It is fundamental in coordinate geometry when determining relationships between points and verifying area-related conditions.

Coordinate Geometry and Line Equations

Understanding line equations is a key component in coordinate geometry. A line in the plane is often expressed in the slope-intercept form or other forms, and it defines a linear relationship between the x and y coordinates of points lying on the line. This concept is crucial when a vertex of a triangle or any other geometric figure is constrained to lie on a particular line, ensuring that the point satisfies a given linear equation.

Determining Points That Satisfy Geometric Constraints

This involves the method of finding or verifying the coordinates of a point that meets multiple conditions, such as lying on a given line while also resulting in a triangle with a specified area when combined with other fixed points. This concept connects the algebraic condition from the area formula with the geometric condition provided by a line equation, enabling the selection or validation of points that satisfy all given constraints.

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