Chapter 4: Q14. (page 186)

Find the area of the triangle shown at the right.

Short Answer

Expert verified

Area of the triangle is 26square units.

Step by step solution

- Define area of triangle

The area of triangle having vertices at $(a, b)$ , $(c, d)$ and $(e, f)$ is $|A|$ , where $A = \frac{1}{2} |\begin{matrix} a & b & 1 \\ c & d & 1 \\ e & f & 1 \end{matrix}|$ . Here vertices of the triangle are $(a, b) = (5, 4)$ , $(c, d) = (- 3, - 2)$ and $(e, f) = (3, - 4)$ . So, substitute 5 for a, 4 for b, -3 for c, -2 for d, 3 for e and -4 for f into the expression $A = \frac{1}{2} |\begin{matrix} a & b & 1 \\ c & d & 1 \\ e & f & 1 \end{matrix}|$ .

$A = \frac{1}{2} |\begin{matrix} 5 & 4 & 1 \\ - 3 & - 2 & 1 \\ 3 & - 4 & 1 \end{matrix}|$

- Find the determinant

Find the determinant in this case by using expansion by minors.

$\begin{matrix} A = \frac{1}{2} |\begin{matrix} 5 & 4 & 1 \\ - 3 & - 2 & 1 \\ 3 & - 4 & 1 \end{matrix}| \\ = \frac{1}{2} (5 |\begin{matrix} - 2 & 1 \\ - 4 & 1 \end{matrix}| - 4 |\begin{matrix} - 3 & 1 \\ 3 & 1 \end{matrix}| + |\begin{matrix} - 3 & - 2 \\ 3 & - 4 \end{matrix}|) \end{matrix}$

- Evaluate 2×2 determinants

The determinant of second order matrix is found by calculating the difference of the product of the two diagonals, that is., $|\begin{matrix} a & b \\ c & d \end{matrix}| = a d - b c$ . Apply this definition to find the $2 \times 2$ determinant.

localid="1648217738738" $\begin{matrix} A = \frac{1}{2} (5 [- 2 - (- 4)] - 4 [- 3 - 3] + [- 3 (- 4) - 3 (- 2)]) \\ = \frac{1}{2} (5 [2] - 4 [- 6] + [12 + 6]) \\ = \frac{1}{2} (10 + 24 + 18) \\ = \frac{1}{2} \times 52 \\ = 26 \end{matrix}$