Medians in Triangles


definition
A median of a triangle is a segment joining any vertex of the triangle to the midpoint of the opposite side.
median

medianNot

All triangles have three medians, which, when drawn, will intersect at one point in the interior of the triangle called the centroid.

mediansconcurrent
The centroid of a triangle divides the medians into a 2:1 ratio. The section of the median nearest the vertex is twice as long as the section near the midpoint of the triangle's side. In other words, the length of the median from the vertex to the centroid is 2/3 of its total length.

hintgal
FYI: When three or more lines intersect in a single (common) point, the lines are referred to as being concurrent. The medians of a triangle are concurrent. Find out more about concurrency in the section on Constructions and Concurrency.

theorem
The median to the hypotenuse in a right triangle is equal to half of the hypotenuse. To be discussed in the section on Right Triangles.

Examples:

1. segTRI1a    segTrimath1

Solution:
M is the midpoint
CM = MB
5x - 2 = 3x + 12
2x = 14
x = 7
CM = 33; CB = 66 units

2.   segtri2    segTriMath2 

Solution:
M, N are the midpoints
DM = ME
4x - 10 = 3x + 5
x = 15
FN = 4x + 3 = 63
NE = 63 units

3.   segTriprac3   segTrimath3

Solution:
M, N , P are the midpoints
AP = 12
AQ = 2/3 of AM = 14
QP = 1/3 of CP = 6
Perimeter = 32 units


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Altitudes in Triangles

definition
An altitude of a triangle is a segment from any vertex perpendicular to the line containing the opposite side.
alt1
alt2
alt3
All triangles have three altitudes, which, when drawn, may lie inside the triangle, on the triangle or outside of the triangle.
alt13
The three altitudes in an acute triangle all lie
in the interior of the triangle and intersect inside the triangle.
alt23
Two of the three altitudes in a right triangle are the
legs of the triangle. The 3 altitudes intersect on the triangle.
alt33Two of the three altitudes in an obtuse triangle lie outside of the triangle. The lines containing the 3 altitudes intersect outside the triangle.
hintgal
Altitudes are perpendicular and form right angles. They may, or may NOT, bisect the side to which they are drawn.
Like the medians, the altitudes are also concurrent. When drawn, the lines containing the three altitudes will intersect in one common point, either inside, on, or outside the triangle. The point where the lines containing the altitudes are concurrent is called the orthocenter of the triangle.
altortho
(The prefix "ortho" means "right".)
Examples:

1. altPicEx1   altN4 

Solution:
altitude is perpendicular
ADB is a right angle of 90º.
5x - 15 = 90
5x = 105
x = 21

2.     altPic2   altN5

Solution:
The altitude will give
m
ADC = 90º, giving
mCAD = 35º.
M is
a midpoint so MB = 12.5

3.     altPicEx3  altN6

Solution:
The altitudes will give right ∠ADM,
MBA and ∠MBP.
m
DMA = 60º
m
AMP = 120º (linear pair)
mAMB = 48º (120º- 72º)
mMAB = 42º (180º - (90º + 48º))

 


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Angle Bisectors in a Triangle

definition
An angle bisector is a ray from the vertex of the angle into the interior of the angle forming two congruent angles.
abacuteA
All triangles have three angle bisectors. The angle bisectors are concurrent in the interior of the triangle.
abacute
The point of concurrency is called the incenter, and is the center of an inscribed circle within the triangle. This fact is important when doing the construction of an inscribed circle in a triangle.

theorem
An angle bisector is equidistant from the sides of the angle when measured along a segment perpendicular to the sides of the angle.
To be discussed in the section on Constructions and Concurrency.
theorem
The bisector of an angle of a triangle divides the opposite side into segments that are proportional to the adjacent sides.
To be discussed in the section on Similarity.

Examples:

1.
abEX1    altN1

Solution:
mACD = mDCB
2x + 15 = 4x - 5
20 = 2x
x = 10
mACD = mDCB = 35
mACB = 70º

2.   abEx2   altN2

Solution:
mRWT = mTWS
mRWT = 32º
mRTW = 77º (180º in Δ)
m
WTS = 103º (linear pair)
(This could also be done using ∠WTS as an exterior angle for ΔRWT.)

3.   abEx3    altN3

Solution:
mABT = mTBC
mABT = 34º
mAVB = 108º (vertical ∠s)
mBAU = 38º (180º in Δ)


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Perpendicular Bisectors in a Triangle

definition
A perpendicular bisector is a line (or segment or ray) that is perpendicular to a side of the triangle and also bisects that side of the triangle by intersecting the side at its midpoint.
The perpendicular bisector may, or may NOT, pass through the vertex of the triangle.
pbEX1
All triangles have perpendicular bisectors of their three sides. The perpendicular bisectors are concurrent, either inside, on, or outside the triangle.
pbacute
The point of concurrency is called the circumcenter, and is the center of a circumscribed circle about the triangle. This fact is important when doing the construction of a circumscribed circle about a triangle.

theorem
The perpendicular bisector of a line segment is the set of all points that are equidistant from its endpoints. To be discussed in the sections on Parallels and Perpendiculars and on Constructions.

Examples:

1. pbEX1a bisN1

Solution:
AD = DC
AD = 9
mAED and mCDE = 90º
mA = 60º

2.   pbEX2  pbEXmath2   

Solution:
PY = YT
5a + 5 = 6a - 1
a = 6
AY = 50

3.   pbEX3   pbEX3

Solution:
BE = EC = 12
DEC right ∠
DC = 13 (Pyth. Thm)
AC = 27




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