Topic: Determinants - Solved Examples – Triangle Area using Determinants
Solved Example 1
Find the area of the triangle whose vertices are (3, -2), (4, 1), and (-1, 3).
- Given coordinates of vertices: A(3, -2), B(4, 1), and C(-1, 3).
- We need to find the area of triangle ABC.
- Area of a triangle using determinants: ( \text{Area} = \frac{1}{2} \left| \begin{array}{ccc} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{array} \right| )
- Plugging the values, we have ( \text{Area} = \frac{1}{2} \left| \begin{array}{ccc} 3 & -2 & 1 \ 4 & 1 & 1 \ -1 & 3 & 1 \end{array} \right| )
Solved Example 1 (contd.)
- Expanding along the first row, we get ( \text{Area} = \frac{1}{2} (3 \times 1 + (-2) \times (-1) + 1 \times 3) - (1 \times 1 + 1 \times 3 + 4 \times (-2)) )
- Simplifying this, we obtain ( \text{Area} = \frac{1}{2} (3 + 2 + 3 + 4 - 1 - 6) )
- Further computation yields ( \text{Area} = \frac{1}{2} (5) = 2.5 ) square units.
- Therefore, the area of triangle ABC is 2.5 square units.
Solved Example 2
Find the area of the triangle formed by the points (2, 1), (5, 7), and (-3, 4).
- Given coordinates of vertices: A(2, 1), B(5, 7), and C(-3, 4).
- We need to find the area of triangle ABC.
- Area of a triangle using determinants: ( \text{Area} = \frac{1}{2} \left| \begin{array}{ccc} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{array} \right| )
- Plugging the values, we have ( \text{Area} = \frac{1}{2} \left| \begin{array}{ccc} 2 & 1 & 1 \ 5 & 7 & 1 \ -3 & 4 & 1 \end{array} \right| )
Solved Example 2 (contd.)
- Expanding along the first row, we get ( \text{Area} = \frac{1}{2} (2 \times 7 + 1 \times 1 + 1 \times 4) - (4 \times 7 + 1 \times (-3) + 2 \times 5) )
- Simplifying this, we obtain ( \text{Area} = \frac{1}{2} (14 + 1 + 4 - 28 - 3 - 10) )
- Further computation yields ( \text{Area} = \frac{1}{2} (-22) = -11 ) square units.
- Therefore, the area of triangle ABC is -11 square units.
Solved Example 3
Find the area of the triangle formed by the points (-1, 3), (2, -1), and (5, 0).
- Given coordinates of vertices: A(-1, 3), B(2, -1), and C(5, 0).
- We need to find the area of triangle ABC.
- Area of a triangle using determinants: ( \text{Area} = \frac{1}{2} \left| \begin{array}{ccc} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{array} \right| )
- Plugging the values, we have ( \text{Area} = \frac{1}{2} \left| \begin{array}{ccc} -1 & 3 & 1 \ 2 & -1 & 1 \ 5 & 0 & 1 \end{array} \right| )
Solved Example 3 (contd.)
- Expanding along the first row, we get ( \text{Area} = \frac{1}{2} ((-1) \times (-1) + 3 \times 1 + 1 \times 0) - (3 \times (-1) + 1 \times 5 + (-1) \times 2) )
- Simplifying this, we obtain ( \text{Area} = \frac{1}{2} (1 + 3) - ((-3) + 5 + (-2)) )
- Further computation yields ( \text{Area} = \frac{1}{2} (4) = 2 ) square units.
- Therefore, the area of triangle ABC is 2 square units.
Solved Example 4
Find the area of the triangle formed by the points (1, 2), (-3, -1), and (-3, 6).
- Given coordinates of vertices: A(1, 2), B(-3, -1), and C(-3, 6).
- We need to find the area of triangle ABC.
- Area of a triangle using determinants: ( \text{Area} = \frac{1}{2} \left| \begin{array}{ccc} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{array} \right| )
- Plugging the values, we have ( \text{Area} = \frac{1}{2} \left| \begin{array}{ccc} 1 & 2 & 1 \ -3 & -1 & 1 \ -3 & 6 & 1 \end{array} \right| )
Solved Example 4 (contd.)
- Expanding along the first row, we get ( \text{Area} = \frac{1}{2} (1 \times (-1) + 2 \times 1 + 1 \times 6) - ((-3) \times (-1) + 1 \times (-3) + (-3) \times 2) )
- Simplifying this, we obtain ( \text{Area} = \frac{1}{2} (-1 + 2 + 6) - (3 - 3 - 6) )
- Further computation yields ( \text{Area} = \frac{1}{2} (7) = 3.5 ) square units.
- Therefore, the area of triangle ABC is 3.5 square units.
Summary
- Area of a triangle using determinants: ( \text{Area} = \frac{1}{2} \left| \begin{array}{ccc} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{array} \right| )
- Determinants can be used to find the area of any triangle given the coordinates of its vertices.
- Practice solving additional examples to enhance your understanding of this concept.