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Medians, altitudes, and perpendicular bisectors
May 1, 2008
A little vocabulary review
• Scalene triangle:________sides are congruent
• Isosceles triangle:________sides are congruent
• Equilateral triangle:________sides are congruent
More vocabulary review
• Acute: ______ acute angle(s)• Obtuse: ______ obtuse angle(s)• Right: ______ right angle(s)• Equiangular: ______ angles are
congruent
Find these measures.
• Find the missing values.
x
x x
x
35
And these…
• Find the missing values.
yx
mDCA = 155
mABC = 72
B
AC
D
What does it mean to be congruent?
• When two figures have the same shape and size, they are called congruent.
• What kinds of congruent triangles are there?
What are the parts of an isosceles triangle?
• Base• Legs• Vertex angle• Base angles
What is a median?
• A median of a triangle is a segment from a vertex to the midpoint of the opposite side.
What is an altitude?
• An altitude of a triangle is the perpendicular segment from a vertex to a line that contains the opposite side.
More about altitudes
• In a right triangle, two of the altitudes are part of the triangle.
• Since the two legs are perpendicular to each other then each leg is also an altitude of the triangle.
• The third altitude is inside the triangle.
Even more about altitudes
• In an obtuse triangle, two of the altitudes are actually outside the triangle.
• This is why the definition states that the altitude is the perpendicular segment from a vertex to the line that contains the opposite side.
What is a perpendicular bisector?
• A perpendicular bisector is a line (or ray or segment) that is perpendicular to a segment at its midpoint.
What are concurrent line?
• Concurrent lines are two or more lines that intersect in one point.
Concurrent medians
• The medians are concurrent at a point called the centroid. The length of the segment from a vertex to the centroid is always 2 /3 the length of the entire median.
Concurrent altitudes
• The lines that contain the altitudes are concurrent at a point called the orthocenter.
Concurrent perpendicular bisectors
• The perpendicular bisectors of the sides are concurrent at a point called the circumcenter.