Answer:
Slope of the other line = 3
Step-by-step explanation:
When two lines intersect at a point, the angles between the lines can be expressed in terms of the slope of the two lines.
[tex]\boxed{\bf Tan \ \theta = \dfrac{m_2-m_1}{1+m_1*m_2}}\\\\\\Here, \ m_1 \ and \ m_2 \ are \ the \ slope \ of \ the \ two \ lines.[/tex]
[tex]\sf \theta = \dfrac{\pi }{4} \ and \ m_1 = \dfrac{1}{2}\\\\\\We \ know \ that \ tan \ \dfrac{\pi }{4} = 1[/tex]
[tex]\sf tan \ \dfrac{\pi }{4}=\dfrac{m_2 - \frac{1}{2}}{1+m_2 *\frac{1}{2}}[/tex]
[tex]\sf 1 = \dfrac{m_2 - \frac{1}{2}}{1+m_2*\frac{1}{2}}\\\\\\Cross \ multiply, \\\\1 + \dfrac{m_2}{2}=m_2 - \dfrac{1}{2}\\\\\\Multiply \ the \ entire \ equation \ by 2,\\2 + m_2 =2m_2 - 1\\\\~~~2 + 1 = 2m_2 - m_2\\\\~~~~~~~~3 =m_2[/tex]
[tex]\boxed{ \sf m_2 = 3}[/tex]
To find the slope of the second line when the angle between two lines is π/4 and the slope of one is 1/2, one can use the tangent function and the relation between the slopes of intersecting lines. The possible slopes for the second line are 2 or -1.
The question is asking to find the slope of the second line given that the angle between two lines is rac{ ext{π}}{4} and the slope of one of the lines is rac{1}{2}.
When two lines intersect, the angle between them, given by θ, relates to the slopes of the lines, m1 and m2, through the formula:
tan(θ) = |(m1 - m2) / (1 + m1*m2)|
Since we know the angle θ is rac{ ext{π}}{4} and tan(rac{ ext{π}}{4}) is 1, and one slope m1 is rac{1}{2}, we can find the slope of the other line, m2, by rearranging the formula:
1 = |(rac{1}{2} - m2) / (1 + rac{1}{2}*m2)|
After solving the equation, we find that there are two possible slopes for the second line, m2 = 2 or m2 = -1.
