Determine the acute angle between the lines 3x - 7y = 21 and 5x + 3y = 15.
angle between two lines if we know the slopes, m1 and m2
tanØ = |(m1 - m2)/(1 + m1m2)|
where Ø is the acute angle
slope of first line = 3/7
slope of 2nd line = -5/3
tanØ = (3/7 + 5/3)/(1 - 5/7)
= (44/21) / (-2/7)
= -22/3
Ø = 180 - 82.23° = 97.77°
or, without the above formula
m1 = 3/7
so the first line makes an angle of 23.199° with the x-axis
m2 = -5/3
so the 2nd line makes an angle of 120.964° with the x-axis
so the angle between them
= 120.964 - 23.199
= 97.77°
However, they wanted the acute angle. So, Sylvia, now it's up to you...
To determine the acute angle between two lines, we need to find the slope of each line and then use the formula:
m₁ = slope of the first line
m₂ = slope of the second line
The formula to find the acute angle between two lines is:
θ = | arctan((m₂ - m₁) / (1 + m₁ * m₂)) |
Now, let's find the slopes of the two lines:
Line 1: 3x - 7y = 21
Rearranging the equation in slope-intercept form (y = mx + b), we get:
-7y = -3x + 21
y = (3/7)x - 3
So, the slope of the first line (m₁) = 3/7
Line 2: 5x + 3y = 15
Rearranging the equation in slope-intercept form (y = mx + b), we get:
3y = -5x + 15
y = (-5/3)x + 5
So, the slope of the second line (m₂) = -5/3
Now, let's substitute the values of m₁ and m₂ into the formula to find the acute angle (θ):
θ = | arctan((-5/3) - (3/7)) / (1 + (3/7) * (-5/3)) |
θ = | arctan((-35/21) - (9/7)) / (1 - (15/21)) |
θ = | arctan(-56/21) / (1 - 15/21) |
θ = | arctan(-8/3) / (6/21) |
θ = | arctan(-8/3) / (2/7) |
θ = | arctan(-8/3) / (2/7) |
θ ≈ | 0.8948 |
Now, taking the absolute value since an angle cannot be negative:
θ ≈ 0.8948
Therefore, the acute angle between the lines 3x - 7y = 21 and 5x + 3y = 15 is approximately 0.8948 radians.
To determine the acute angle between two lines, we need to find the slopes of the lines and then use the formula for the angle between two lines.
Step 1: Rewrite the given equations in slope-intercept form (y = mx + b):
3x - 7y = 21 => 7y = 3x - 21 => y = (3/7)x - 3
5x + 3y = 15 => 3y = -5x + 15 => y = (-5/3)x + 5
Step 2: Compare the slopes of the two lines.
The slope of the first line is (3/7) and the slope of the second line is (-5/3).
Step 3: Apply the formula to find the acute angle between the lines.
The formula for the acute angle between two lines with slopes m1 and m2 is given by:
tan(theta) = |(m1 - m2) / (1 + m1 * m2)|
Substituting the slopes into the formula, we have:
tan(theta) = |((3/7) - (-5/3)) / (1 + (3/7) * (-5/3))|
Simplifying further:
tan(theta) = |(9/21 + 35/21) / (1 - 15/21)| = |(44/21) / (6/21)| = |(44/6)| = 22/3
Step 4: Calculate the value of theta using the inverse tangent function:
theta = arctan(22/3) ≈ 82.91 degrees
Therefore, the acute angle between the lines 3x - 7y = 21 and 5x + 3y = 15 is approximately 82.91 degrees.