Question

what is the area of the triangle with vertices $(1,5)$, $(4, - 1)$, and $(6, - 4)$?
$a=\frac{1}{2}|\det a|$, where $a = \

$$\begin{bmatrix}x_1&y_1&1\\\\x_2&y_2&1\\\\x_3&y_3&1\\end{bmatrix}$$

$
(1 point)
$\bigcirc 10.5$
$\bigcirc 1.5$
$\bigcirc 2.5$
$\bigcirc 11.5$

Explanation:

Step1: Substitute the coordinates into the matrix

Given vertices \((x_1,y_1)=(1,5)\), \((x_2,y_2)=(4, - 1)\), \((x_3,y_3)=(6,-4)\).
The matrix \(A=

$$\begin{bmatrix}1&5&1\\4&-1&1\\6&-4&1\end{bmatrix}$$

\)

Step2: Calculate the determinant of the matrix

$$
LATEXBLOCK1
$$

Step3: Calculate the area of the triangle

Using the formula \(A=\frac{1}{2}|\det(A)|\), substitute \(\det(A) = 3\)
\(A=\frac{1}{2}\times|3|=\frac{3}{2}=1.5\)

Answer:

\(1.5\)