In the diagram, P, R(3; 5), S(-3; -7) and T(-5; k) are vertices of trapezium PRST and PT || RS - NSC Mathematics - Question 3 - 2019 - Paper 2

To calculate the gradient (m) of line RS, we use the formula for slope:

m = rac{y_2 - y_1}{x_2 - x_1}

For points R(3; 5) and S(-3; -7):

$m_{RS} = \frac{-7 - 5}{-3 - 3} = \frac{-12}{-6} = 2$

We know that:

$m_{RS} = 2$

Using the tangent formula:

$\tan(θ) = \frac{m_{RS} - m_{PR}}{1 + m_{RS} * m_{PR}}$ To find θ:

$θ = \tan^{-1}(2)$

Using a calculator, we find:

$θ ≈ 63.43°$

To find the coordinates of point D, we need to identify where line RS cuts the y-axis. We can use the equation of line RS with x=0 to find the y-coordinate:

Substituting x = 0:

$y = 2(0) - 1 = -1$

Thus, D(0; -1).

Given the length of TS and knowing the coordinates of T(-5; k) and S(-3; -7), we calculate the distance:

Using the distance formula:

$TS = \sqrt{(-5 - (-3))^2 + (k - (-7))^2} = 2/\sqrt{5}$

Solving gives:

$2 = 2/\sqrt{5}$

Squaring both sides and simplifying results in: $k = -3$

To find the coordinates of N, we use the midpoint theorem. The coordinates of N are determined by the vertices of the parallelogram:

$N = T + (D - S)$ Where T(-5; k) and since D = (0; -1) and S = (-3; -7), we can find:

$(N_x, N_y) = (0 + 2, -1 + 6) = (2, 5)$. So coordinates of N are (2, 5).

To find the size of angle R'D'R', we can use the cosine rule. First, we calculate the distances using the coordinates of R, D and R' Given points: R(3; 5) D(0; -1) After reflection, calculate sizes using the cosine rule:

$R'D'R' = R'D^2 + D'R^2 - 2(R'D')(D'R)cos(θ)$ Evaluate using values to find the size after reflection about the y-axis.