Question

Use determinants to find the area of the parallelogram shown below

Answer 1

30

Explanation:

To find the determinant of a parallelogram given points (a, b), (c, d), and (e, f), we can use

`\left[\begin{array}{ccc}a&b&1\\c&d&1\\e&f&1\end{array}\right]` and calculate the determinant of that. Three points on the parallelogram are (-1,1), (-1, -5), and (4, 5). Plugging these into the matrix, we get

`\left[\begin{array}{ccc}-1&1&1\\-1&-5&1\\4&5&1\end{array}\right]`. The determinant is equal to

`-1 *det \left[\begin{array}{ccc}-5&1\\5&1\end{array}\right] \\- 1 * det \left[\begin{array}{ccc}-1&1\\4&1\end{array}\right] \\\\+ 1 * det \left[\begin{array}{ccc}-1&-5\\4&5\end{array}\right] \\= -1 * (-5*1 - (1*5))- 1 * (-1 * 1 - (4*1)) + 1 * (-1 * 5 - (-5*4)) \\= -1 *(-5-5) -1 * (-1 - 4) + 1 * (-5 - (-20))\\= -1 * (-10) -1 * (-5) +1 * (15)\\= 10 + 5 + 15\\=30`