Problem:
What is the degree measure of the acute angle formed by lines with slopes 2 and 1/3?
Answer Choices:
A. 30
B. 37.5
C. 45
D. 52.5
E. 60
Solution:
Consider lines from the origin O to points P(3,1) and Q(β1,3). The lines are perpendicular because their slopes are negative reciprocals of each other. Points P and Q are equidistant from O, so their midpoint lies on the bisector of β POQ. The midpoint is
M=(23+(β1)β,21+3β)=(1,2)
and the line OM has slope 2. Angle β MOP is formed by lines with slopes 2 and 31β and is half of a right angle, so its measure is (C)45ββ.
OR
Define points O(0,0),M(1,2), and P(3,1), so that lines OM and OP have the required slopes. Then OM=12+22β=5β,MP=(3β1)2+(2β1)2β=5β, and OP=32+12β=10β.
Furthermore, OP=2ββ OM. It follows that β³OMP is an isosceles right triangle with hypotenuse OP, so β MOP=(C)45ββ.
OR
The specified angle is the difference between angles Ξ± and Ξ² whose tangents are 2 and 31β, respectively. The Tangent Difference Formula gives
