In die diagram is A(4; 2), B(6; -4) en C(-2; -3) hoekpunte van AABC - NSC Mathematics - Question 3 - 2022 - Paper 2

The angle α can be calculated using the tangent function:

$\tan(\alpha) = m_{AB} = -3 \Rightarrow \alpha = \tan^{-1}(-3) \approx 108,43°$

Point T is the midpoint of line CB. The coordinates can be calculated using the midpoint formula:

$T = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \Rightarrow \left(\frac{6 + (-2)}{2}, \frac{-4 + (-3)}{2}\right) \Rightarrow T(2; -3,5)$

To find the coordinates of point S where lines AC and DT intersect, we first need to determine the equations of each line. After computing the intersections:

The coordinates of S are approximately ( S(0; -\frac{4}{3}) ).

The gradient of line CD is calculated based on the coordinates of points C and D. Formulating the line equation results in:

The equation for line CD can be expressed as: [ y = -3x + 9 ]

The size of angle ∠DCA can be determined using the known gradients and the formula for the angle between two lines:

$\tan(θ) = \frac{|m_1 - m_2|}{1 + m_1 m_2}$

Thus, we calculate this angle to find ∠DCA value.

The area of polygon POSC can be calculated using the formula for the area of a polygon based on its vertices:

$\text{Area} = \frac{1}{2} | x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) |$
This leads to the final area calculation being 5,83 square units.