The straight lines x+y-4=0, 3x+y-4=0 and x+3y-4=0 form a triangle which is:
Question
The straight lines x+y-4=0, 3x+y-4=0 and x+3y-4=0 form a triangle which is:
Solution
The given lines are x+y-4=0, 3x+y-4=0 and x+3y-4=0.
First, let's find the slopes of these lines.
The slope of a line given by the equation ax+by+c=0 is -a/b.
So, the slopes of the given lines are:
- For x+y-4=0, slope = -1/1 = -1
- For 3x+y-4=0, slope = -3/1 = -3
- For x+3y-4=0, slope = -1/3
Now, let's find the angles between these lines using the formula for the angle θ between two lines with slopes m1 and m2:
tan(θ) = |(m1 - m2) / (1 + m1*m2)|
- Angle between lines 1 and 2: tan(θ) = |(-1 - (-3)) / (1 + -1*-3)| = 2/4 = 0.5, so θ = arctan(0.5) ≈ 26.57°
- Angle between lines 2 and 3: tan(θ) = |(-3 - (-1/3)) / (1 + -3*-1/3)| = 2.67/2 = 1.335, so θ = arctan(1.335) ≈ 53.13°
- Angle between lines 1 and 3: tan(θ) = |(-1 - (-1/3)) / (1 + -1*-1/3)| = 0.67/2 = 0.335, so θ = arctan(0.335) ≈ 18.43°
Since all angles are less than 90°, the triangle formed by these lines is acute.
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