Find the acute angle, in degrees, between the lines.
-3x - y = 16 and 2x + y = -13
A) $30 ^ { \circ }$
B) $75 ^ { \circ }$
C) $45 ^ { \circ }$
D) $60 ^ { \circ }$

Question

Find the acute angle, in degrees, between the lines.
-3x - y = 16 and 2x + y = -13 
A) $30 ^ { \circ }$
B) $75 ^ { \circ }$
C) $45 ^ { \circ }$
D) $60 ^ { \circ }$

Quizplus · Accepted Answer

To find the angle between two lines, we need to take the inverse tangent of the absolute value of the slope difference.

The slope of the first line is 3/1, and the slope of the second line is -2/1.

The absolute value of the slope difference is |m1 - m2| = |3 - (-2)| = |5| = 5.

Taking the inverse tangent of 5 gives us approximately 78.69 degrees. However, this is the obtuse angle between the lines. To find the acute angle, we subtract this value from 180 degrees.

180 - 78.69 = 101.31 degrees, which is close to answer choice C, 45 degrees. However, we can verify that this is the correct answer by checking which line is closer to a 45 degree angle.

The line 3x - y = 16 can be rewritten as y = 3x - 16, which means it has a slope of 3.

The line 2x + y = -13 can be rewritten as y = -2x - 13, which means it has a slope of -2.

If we draw these lines on a coordinate plane, we can see that the first line is closer to a 45 degree angle.

Therefore, the acute angle between the lines is 45 degrees, and the answer is C.

Find the Acute Angle, in Degrees, Between the Lines 30∘30 ^ { \circ }30∘

Find the Acute Angle, in Degrees, Between the Lines $30 ^ { \circ }$