The diagram below shows a side view of a slanted ladder KL against a vertical wall KZ - NSC Technical Mathematics - Question 1 - 2021 - Paper 2

The length of KL can be calculated using the distance formula:

$KL = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substituting the coordinates of K(1; 7) and L(-3; -1): $KL = \sqrt{(1 - (-3))^2 + (7 - (-1))^2} = \sqrt{(4)^2 + (8)^2} = \sqrt{16 + 64} = \sqrt{80} \approx 8.94$

The midpoint M of KL can be found using the midpoint formula:

$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ Substituting K(1; 7) and L(-3; -1): $M = \left( \frac{1 + (-3)}{2}, \frac{7 + (-1)}{2} \right) = \left( \frac{-2}{2}, \frac{6}{2} \right) = (-1; 3)$

As calculated previously, the gradient of KL is given by: $m_{KL} = \frac{y_2 - y_1}{x_2 - x_1} = 2$

To calculate the angle θ formed by the line KL and the x-axis, we use the formula: $\tan(\theta) = m_{KL} = 2$ Thus, $\theta = \tan^{-1}(2) \approx 63.4°$ (rounded to one decimal place).

The equation of a line can be expressed as: $y - y_1 = m(x - x_1)$ Given the slope m = 2 from above and passing through (-5; 1):

1. Substitute into the equation: $y - 1 = 2(x + 5)$
2. This simplifies to: $y = 2x + 10 + 1 \Rightarrow y = 2x + 11$

To determine if the point (-4; -2) lies on the line defined by the equation previously calculated:

1. Substitute x = -4 into the line's equation: $y = 2(-4) + 11 = -8 + 11 = 3$
2. Since the y-value derived (3) does not equal -2, point (-4; -2) does NOT lie on the line y = 2x + 11.