Question

In Exercises 11 and 12 , find a value of \(y\) such that the triangle with the given vertices has an area of 4 square units. $$ (-4,2),(-3,5),(-1, y) $$

Short answer

After solving the equations as detailed in step 3, we will get the two possible values of \(y\) that will give the triangle an area of 4 square units.

Step-by-step solution

Understanding the Formula

The formula for calculating the area of a triangle given the vertices \((x_1, y_1)\), \((x_2, y_2)\) and \((x_3, y_3)\) in the coordinate plane is \(Area = \frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|\). This formula will be used to calculate the value of \(y\) required for the area to be 4.

Apply the Formula and Solve for \(y\)

Substituting the given vertices \((-4, 2)\), \((-3, 5)\), and \((-1, y)\) into the formula, we get \(4 = \frac{1}{2}|-4(5-y) - 3(y-2) - 1(2-5)|\). Clearing the absolute value and solving for \(y\) will yield the required values of \(y\).

Solve the Equation

We have to solve two scenarios because of the absolute value. First, consider \(4 = \frac{1}{2}|-4(5-y) - 3(y-2) - 1(2-5)|\) where the equation inside the absolute value is positive. Simplify and solve for \(y\). After finding that value, let's consider the scenario for when the equation inside the absolute value is negative. Again, simplify and solve for \(y\).

Key concepts

Algebraic Geometry

Algebraic geometry is a branch of mathematics that studies the solutions to algebraic equations using geometric concepts. In the context of our exercise, this discipline allows us to visualize and solve problems involving the coordinates of points and the shapes they form—like the triangle in our problem.

When we look at the triangle on the coordinate plane formed by the points (-4,2), (-3,5), and (-1, y), we are essentially investigating a geometric representation of an algebraic problem. Using algebraic geometry, we translate the area of this triangle into an equation that can be solved algebraically.

It's important to understand that the coordinates are not just numbers; they represent positions in a two-dimensional plane. Thus by studying their relationships, we can derive properties of the triangle, such as its area.

Absolute Value Equations

Absolute value equations involve expressions within absolute value bars, which denote the distance of a number from zero on the number line, regardless of direction. This means that the absolute value of a number is always non-negative.

In our step by step solution, the absolute value plays a crucial role. The area formula incorporates an absolute value to ensure that the calculated area is positive, as a geometric area cannot be negative. As shown in the solution, when solving for the variable (y), it is necessary to consider both the positive and negative scenarios of the contents inside the absolute value bars.

This leads to two separate equations: one where we equate the expression inside the absolute value bars to a positive 8 (since the area has to be doubled when solving), and one where it is equated to negative 8. Each equation will result in a different potential value for (y), ultimately determining the coordinates that make the triangle's area exactly 4 square units.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, deals with shapes and lines on the coordinate plane. It provides a link between algebra and geometry through graphs of lines and curves. This helps visualize algebraic entities and solve geometric problems algebraically.

In our exercise, we use coordinate geometry to find the area of a triangle with vertices located at specific points on the coordinate plane. By using the given formula, which is a direct application of coordinate geometry principles, we can determine the exact location of the third vertex that will give us a triangle of the desired area.

The formula incorporates the x and y coordinates of the triangle's vertices and effectively uses the rules of coordinate geometry to solve for the unknown y. This is an excellent example of how algebra and geometry can be combined to solve a problem that at first glance may seem purely geometric.