Questions: Challenging Question In the figure, straight lines L1, L2 and L3 intersect at the same point. Denote the acute angle between L2 and L3 by Q (a) Find the inclination of L2 (b) If the slope of L3 is 0.5 , find Q, correct to the nearest degree.

Challenging Question

In the figure, straight lines L1, L2 and L3 intersect at the same point. Denote the acute angle between L2 and L3 by Q
(a) Find the inclination of L2
(b) If the slope of L3 is 0.5 , find Q, correct to the nearest degree.

Transcript text: Challenging Question In the figure, straight lines $L_{1}, L_{2}$ and $L_{3}$ intersect at the same point. Denote the acute angle between $L_{2}$ and $L_{3}$ by $Q$ (a) Find the inclination of $L_{2}$ (b) If the slope of $L_{3}$ is 0.5 , find $Q$, correct to the nearest degree. Solution (a)

Solution

Solution Steps

Step 1: Finding the inclination of $L_2$

The inclination of $L_2$ is the angle it makes with the positive x-axis. In the figure, this is given by $78^\circ + 180^\circ = 258^\circ$. Since the inclination is usually given as an angle between $0^\circ$ and $180^\circ$, and the angle between $L_2$ and the positive x-axis in the other direction is $180^\circ - 78^\circ = 102^\circ$, we take $102^\circ$ as the inclination of $L_2$.

Step 2: Finding the inclination of $L_3$

Let $\theta$ be the inclination of $L_3$. The slope of $L_3$ is given as $0.5$. The slope of a line is equal to the tangent of its inclination. Therefore, $\tan(\theta) = 0.5$. This means $\theta = \arctan(0.5) \approx 26.57^\circ$.

Step 3: Finding the acute angle between $L_2$ and $L_3$

The angle between $L_2$ and $L_3$ is the difference in their inclinations. The inclination of $L_2$ is $102^\circ$, and the inclination of $L_3$ is approximately $26.57^\circ$. The acute angle $\alpha$ between $L_2$ and $L_3$ is therefore $\alpha \approx |102^\circ - 26.57^\circ| = 75.43^\circ \approx 75^\circ$.