a Horizontal and vertical lines are perpendicular.
B
b Perpendicular slopes have a product of - 1.
A
aExample Solution: D(0,3)
B
bExample Solution: C(3, -1)
Practice makes perfect
a Let's begin with illustrating ∠ AOX.
Before moving on, we have to explain what adjacent and complementary angles mean:
Two angles are said to be adjacent when they have a common side and a common vertex.
Complementary angles have measures that add up to 90^(∘).
The common vertex is O but the common side could be either OA or OX. Therefore, we get two possibilities which have been illustrated below. Note that these angles are congruent according to the Congruent Complements Theorem.
Examining the diagram, we see that one ray is vertical along the y-axis. Therefore, it's relatively easy to find the coordinates of a point B that creates an adjacent and complementary angle with ∠ AOX as all we need to do is to pick a point on the y-axis. Why not pick (0,3)?👍
Let's illustrate the complementary angles.
b To find another point C so that OC is a side of a different angle, we have to use the second ray which is perpendicular to OA. To find the perpendicular slope to OA, we first need the rays slope.
Knowing the vertical and horizontal distance between O and A, we can find the rays slope.
m=Δ y/Δ x ⇒ m= 3/1=3
The slope of perpendicular lines are negative reciprocals of each other. This means that the product of their slopes is - 1. With this, we can find the perpendicular slope.
The ray perpendicular to OA has a slope of - 13. With this, we can find a point, C, so that OC is adjacent and complementary to ∠ AOX.
One possible solution is C(3,- 1). However, notice that there are infinitely more solutions and that the coordinates do not necessarily have to be whole numbers. Let's illustrate the complementary angles.
