Problem 11 - College Algebra and Calculus: An Applied Approach

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. $$ (-5,1),(0,2),(-2, y) $$

Step-by-Step Solution

Answer

The value of \(y\) that makes the area of the triangle 4 square units is either \(y = \frac{4}{5}\) or \(y = \frac{12}{5}\).

Step 1: Write down the known vertices and formula

The given vertices are \((-5,1)\), \((0,2)\), and \((-2, y)\), and the formula for the area of a triangle using vertices is \(\frac{1}{2}|x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|\).

Step 2: Substitute the known vertices into the formula

Substitute the given vertices into the formula, it becomes \(\frac{1}{2}|-5(2 - y) + 0(y - 1) - 2(1 - 2)|\).

Step 3: Simplify the formula

Simplify the expression inside the absolute value, this results into, \(\frac{1}{2}|-10 + 5y + 2|\).

Step 4: Set the formula equal to the area

We know that the area of the triangle should equal to 4 square units. So we set the formula equal to 4, which gives us \(\frac{1}{2}|-10 + 5y + 2| = 4\).

Step 5: Solve for \(y\)

Solve the equation \(\frac{1}{2}|-10 + 5y + 2| = 4\), for \(y\), which gives us two possible solutions: \(y = \frac{4}{5}\) or \(y = \frac{12}{5}\)

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This allows us to analyze geometric figures in a numerical way, making it possible to solve problems using algebra. The coordinates are written as pairs

• First element, usually represented as \( x \), called the x-coordinate or abscissa.
• Second element, usually represented as \( y \), called the y-coordinate or ordinate.

Each coordinate pair gives a precise location of a point in the plane.
In the exercise, we deal with a triangle in a coordinate plane defined by three vertices:

• \((-5,1)\)
• \((0,2)\)
• \((-2,y)\)

The goal is to find a specific \( y \)-value so that the triangle formed by these vertices has a set area of 4 square units. Understanding coordinate geometry helps us to effectively utilize the formula for a triangle's area by substituting the given vertex coordinates.

Vertex Formulation

The Vertex Formulation is crucial when calculating the area of a triangle using vertices. The formula needed is \[\text{Area} = \frac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|\] Here,

• \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) are the coordinates of the vertices of the triangle.
• The bars \(| \, \cdot \, |\) indicate that we take the absolute value, ensuring a non-negative area.

Using vertices, we substitute the given points into the formula. For our problem:

• \( x_1 = -5, y_1 = 1 \)
• \( x_2 = 0, y_2 = 2 \)
• \( x_3 = -2, y_3 = y \)

Inserting these values into the expression, simplifies and rearranges it to solve for \( y \). Understanding vertex formulation is essential to find the solution and ensure the specific triangle area requirement is met.

Absolute Value Equations

Absolute value equations like the one used in the area calculation of this triangle are found in many geometric calculations. The area of the triangle is given by this formula: \[\frac{1}{2}|-10 + 5y + 2| = 4\] Using absolute values, we ensure calculations remain non-negative, which is logical since area cannot be negative. When solving such equations, we'll generally have two cases to consider:

• \(-10 + 5y + 2 = 8\)
• \(-10 + 5y + 2 = -8\)

Solve each equation separately to find possible values of \( y \). This is why we end up with two potential solutions for \( y \),

• \( y = \frac{4}{5}\)
• \( y = \frac{12}{5}\)

Both satisfy the area condition. Understanding and solving absolute value equations is crucial in deriving relevant solutions in geometry that satisfy real-world conditions.