Find the angle made by the line joining (4,3) and (-2,-3) with the positive direction of X-axis
Answers
Answer:
Although the solution is smaller, I am giving a proper explanation of the process. For the shorter solution dive down to the end of the answer.
Let us first find the line joining (5, 3) and (-1, -3). The general form of the line is:
y=mx+cy=mx+c , where m is the slope and c is a constant.
The slope is defined as:
m=(y1−y2)/(x1−x2)m=(y1−y2)/(x1−x2) ……. (1)
Here (x1, y1) = (5, 3) and (x2, y2) = (-1, -3)
Substituting the values in the equation (1), we get
m = (3-(-3)) / (5-(-1)) = (3+3) / (5 + 1)
=> m = 6/6 = 1
Hence m = 1, which gives us the equation of the line as:
y=1.x+cy=1.x+c
y = x + c
To get the value of the constant c, we can use either of the given points
lets take (5, 3) for simplicity,
=> 5=3+c5=3+c
=> c=−2c=−2
So the equation of the line is: y = x -2.
The line meets the x-axis at y=0, putting this value in the equation of the line,
we get x = 2.
i.e. (2, 0) is a point on the line.
Now we get a right-angled triangle ABC between three points A(2, 0) , B(5, 3) and C(5, 0).
Using this triangle we shall find the angle between the line and the positive x-axis.
Our line is AB, and x-axis is AC. BC is perpendicular to x-axis. The want to find the angle between AB and AC, let it be p.
Since it is a right-angled triangle, we can say that:
tanp=height/base=BC/ACtanp=height/base=BC/AC
=>tanp=3/3=1=>tanp=3/3=1
=>tanp=tan45deg=>tanp=tan45deg
=>p=45deg=>p=45deg
Hence the angle is 45deg. our answer.
Short Solution: if you analyse closely, BC/AC is nothing but the slope m.
Hence, tanp=mtanp=m
=>p=arctanm=>p=arctanm
=>p=arctan1=45deg=>p=arctan1=45deg , our answer.
p=45degp=45deg
Thank you for reading.
For better understanding of the concepts:
How use the slope formula and find the slope of a line, whether the Slope is positive, negative or undefined.