15
Perpendicular lines
Textbook: pp. 38–39
GoalKnow perpendicular lines and distance from a point to a line.
New words
perpendicular · perpendikulyaroblique segment · og‘mafoot of the perpendicular · perpendikulyar asosidistance from a point to a line · nuqtadan to‘g‘ri chiziqqacha masofa
Explanation
Two lines that form a right angle where they meet are called perpendicular; we write a ⊥ b. Perpendicular lines form four right angles. Theorem: through any point of a line exactly one line perpendicular to it can be drawn (proof: by the axiom on laying off angles, a 90° angle can be laid off in only one way). If segment AB is perpendicular to line c, it is the perpendicular dropped from A to c, and B is its foot. A segment that is not perpendicular is called oblique. The distance from A to line c is the length of that perpendicular; it is shorter than any oblique segment from the same point (proved later).
Worked examples
If a ⊥ b and one angle at the intersection is 90°, the other three are 90° too.
From A to line c the perpendicular AB is 6 cm and an oblique AC is 9 cm; the distance is the shorter, 6 cm.
Class activity
“Set square”: with a set square draw a perpendicular to a line and the distance to it from a point A.
Practice
1
a ⊥ b. How many right angles appear at the intersection?
4
2
The perpendicular from a point to a line is 7 cm and an oblique is 10 cm. What is the distance (cm)?
7
3
Perpendicular lines meet; the four angles add up to how many degrees?
360
4
Why is there only one perpendicular to a line through a point on it?
Because a 90° angle can be laid on a ray into a half-plane in only one way.