The area of the triangle formed by the point \(\left(a, a^{2}\right),\left(b, b^{2}\right)\) \(\left(\mathrm{c}, \mathrm{c}^{2}\right)\) is \((\mathrm{a}, \mathrm{b}, \mathrm{c}\) are three consecutive odd integers) (a) \((1 / 2)(\mathrm{a}-\mathrm{b})(\mathrm{b}-\mathrm{c}) \mathrm{sq}\) unit (b) 8 sq unit (c) \(16 \mathrm{sq}\) unit (d) \((1 / 2)(\mathrm{a}-\mathrm{b})(\mathrm{b}-\mathrm{c})(\mathrm{a}+\mathrm{b}+\mathrm{c}) \mathrm{sq}\) unit

Calculating the area of a triangle can seem daunting, especially when dealing with coordinate geometry. However, the determinant formula for area simplifies this process immensely. This formula is particularly useful when the coordinates of the triangle's vertices are given. It involves placing the coordinates into a determinant and evaluating it, which provides a direct way to compute the area.

The basic determinant formula is given by:
\[Area = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|\]
By substituting the specific coordinates into this formula, and sometimes simplifying the resulting expression, we obtain the area of the triangle. The absolute value is necessary because the determinant may yield a negative number, but the area must always be positive. Through this method, the complexities of geometry become more manageable.

Consecutive odd integers are numbers in a sequence where each number is two units greater than the previous odd number. For instance, 1, 3, and 5 are consecutive odd integers. In the context of geometry problems, these integers are often used to define the spatial configuration of points or lines.

When given consecutive odd integers as coordinates, it's important to remember their specific properties:

• There is a specific distance between each integer, namely 2.
• This regular spacing can sometimes help to simplify algebraic expressions.
• Using them thoughtfully can uncover patterns that may aid in solving geometric problems.

For example, odd integers used in the vertices of a triangle could lead to simplifications when finding the area with the determinant formula. These integer properties can result in interesting geometric relationships.

Coordinate geometry, also known as analytic geometry, is a powerful branch of mathematics that allows us to represent and analyze geometrical shapes and relationships through algebra. It combines algebraic and geometric techniques, enabling problems to be solved with equations and graphs on the coordinate plane.

In coordinate geometry, every point, line, and shape can be described using coordinates – a pair of numbers denoting their position. It is beneficial to have a solid understanding of the coordinate plane, including the x and y-axes, and how to plot points: using ordered pairs \((x, y)\). When solving problems, like calculating the area of a triangle, knowing how to use formulas specific to coordinate systems is crucial.

In the realm of geometry, the absolute value represents the magnitude of a number, disregarding its sign. This concept is vital because distances and measurements in geometry are inherently nonnegative. When using the determinant formula for the area of a triangle in coordinate geometry, the absolute value ensures that the computed area is a positive number, as a negative area does not make sense in a physical context.

Therefore, when you see \(|expression|\), it signifies that whatever the result is inside the absolute value, you will only consider its positive magnitude. For instance, when you find the area of a triangle using determinants, even if the determinant gives a negative result, the absolute value will convert it into a positive area. This aligns with our intuitive understanding that the size or extent of a geometric shape cannot be negative.