21
Key elements of a triangle: median, altitude and bisector
Textbook: pp. 56–57
GoalTell apart and draw a median, altitude and bisector.
New words
median · medianaaltitude · balandlikangle bisector of a triangle · uchburchak bissektrisasifoot of the perpendicular · perpendikulyar asosi
Explanation
A segment joining a vertex to the midpoint of the opposite side is a median. The segment from a vertex to the opposite side that halves the angle at that vertex is the bisector of the triangle. The perpendicular dropped from a vertex to the line containing the opposite side is the altitude; its foot is often called H. A triangle has three vertices, so it has three medians, three altitudes and three bisectors. In an obtuse triangle the altitudes from the acute-angle vertices can lie outside the triangle, because the perpendicular falls on the extension of the opposite side.
Worked examples
In △ABC, BM is a median: M is the midpoint of AC; if AC = 14 then AM = MC = 7.
In △ABC, BL is the bisector; if ∠B = 70° then ∠ABL = ∠LBC = 35°.
Class activity
“Three lines”: make a paper triangle (an adult cuts it), fold to find a median, use a set square for an altitude.
Practice
1
In △ABC, AC = 18 cm and BM is a median. Find AM (cm).
9
2
In △ABC, BL is a bisector and ∠B = 84°. Find ∠ABL (degrees).
42
3
How many medians does a triangle have?
3
4
Why do some altitudes of an obtuse triangle lie outside?
The perpendicular lands on the extension of the opposite side, not on the side itself.