Question

question 3 in the diagram below, abcd is a kite. it is given that $hat{b}_{1}=55.31^{circ}$. bc is two times the length of ad. determine the gradient of line bc. 3.1 determine the size of $\theta$. 3.2

Explanation:

Step1: Recall gradient formula

The gradient formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $B(2,4)$ and $C(5,0)$, we have $x_1 = 2,y_1=4,x_2 = 5,y_2 = 0$.

Step2: Calculate the gradient

Substitute the values into the formula: $m=\frac{0 - 4}{5 - 2}=\frac{-4}{3}$.

Step3: Recall the relationship between gradient and angle

The gradient $m=\tan\theta$. We know $m =-\frac{4}{3}$, so $\tan\theta=-\frac{4}{3}$. Since the line has a negative - slope, the angle $\theta$ is in the second or fourth quadrant. We find $\theta=\arctan(-\frac{4}{3})$. Using a calculator, $\theta\approx - 53.13^{\circ}+180^{\circ}n$, $n\in\mathbb{Z}$. In the context of the problem (assuming the angle is measured from the positive $x$ - axis), we take the principal value of the angle in the range $0^{\circ}\leq\theta<360^{\circ}$, so $\theta\approx126.87^{\circ}$.

Answer:

3.1. The gradient of line $BC$ is $-\frac{4}{3}$.
3.2. The size of $\theta$ is approximately $126.87^{\circ}$.