4.3 Q-31

Question Statement

Find the area of the triangular region whose vertices are:

$$A(5, 3), , B(-2, 2), , C(4, 2)$$

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Background and Explanation

To find the area of a triangle when its vertices are given, we use the determinant formula for the area of a triangle. This method calculates the area using the coordinates of the vertices:

$$\Delta = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$$

Where:

• $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are the coordinates of the vertices.

This formula is derived from the determinant of a matrix representing the triangle’s vertices. It ensures a positive area regardless of vertex ordering.

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Solution

Step 1: Write down the given vertices

The vertices of the triangle are:

$$A(5, 3), , B(-2, 2), , C(4, 2)$$

Assign:

$$(x_1, y_1) = (5, 3), \quad (x_2, y_2) = (-2, 2), \quad (x_3, y_3) = (4, 2)$$

Step 2: Apply the determinant formula

Substitute the coordinates into the formula:

$$\Delta = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$$

Step 3: Simplify each term

First term:

$$x_1(y_2 - y_3) = 5(2 - 2) = 5(0) = 0$$

Second term:

$$x_2(y_3 - y_1) = -2(2 - 3) = -2(-1) = 2$$

Third term:

$$x_3(y_1 - y_2) = 4(3 - 2) = 4(1) = 4$$

Step 4: Add the terms and take the absolute value

Add the results:

$$0 + 2 + 4 = 6$$

Take the absolute value:

$$|6| = 6$$

Step 5: Divide by 2

Divide by 2 to get the area:

$$\Delta = \frac{1}{2} \cdot 6 = 3 , \text{square units}$$

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Key Formulas or Methods Used

• Determinant formula for the area of a triangle:

$$\Delta = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$$

• Simplification of algebraic expressions.

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Summary of Steps

1. Identify the vertices of the triangle and assign $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$.
2. Substitute the coordinates into the determinant formula:

$$\Delta = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$$

3. Simplify each term in the formula.
4. Add the terms, take the absolute value, and divide by 2.
5. The area of the triangle is $3 , \text{square units}$.