CH 2.3 Shadows and Proportions Proportions and Indirect Measurement
Three Theorems Involving Proportions Section 8.5.
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Transcript of Three Theorems Involving Proportions Section 8.5.
Three Theorems Three Theorems Involving ProportionsInvolving Proportions
Section 8.5Section 8.5
A T
F
B N
Given: AT ║ BN
FA = 12, FT = 15
AT = 14, AB = 8
Find: length BN & NT
Prove similar triangles firstProve similar triangles first
base
side
14
12
BN
20=
14
15
31
23
15 TN=
BN = 23 BN = 23 11//33
TN = 10TN = 10
Theorem 65: If a line is ║to one side of a Δ and intersects the other 2 sides, it divides those 2 sides proportionally. (Side splitter theorem)
A B
C
E
D
FG
Given: AB CD EF
Conclusion:
AC
CE
BD
DF=
Draw AF
In Δ EAF, CEAC
= GF
AG
Bottom
Top
In Δ ABF,
GF
AG= DF
BD
Substitute BD for AG and DF for GF,
substitution or transitive
CE
AC= DF
BD
Theorem 66: If three or more parallel lines are intersected by two transversals, the parallel lines divide the transversals proportionally.
DC
B
AGiven: ∆ ABD
AC bisects <BAD
Prove:
BC
CD
AB
AD=
Set up ProportionsSet up Proportions
AB
BC
AD
CD= Rightside
Rightbase
leftside
Leftbase =
BC • AD = AB • CD Means Extreme
CD
BC
Therefore:
=AD
ABRightside
Leftside
Rightbase
Leftbase=
Theorem 67: If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides. (Angle Bisector Theorem)
a
G
ZS
Y
I
P
X
W
b
c
d
2
3
4
Given: a, b, c and d are parallel lines, Lengths given, WZ = 15
Find: WX, XY and YZ
According to Theorem 66, the ratio of WX:XY:YZ is 2:3:4.Therefore, let WX = 2x, XY = 3x and YZ = 4x2x + 3x + 4x = 159x = 15x = 5/3
WX = 10/3 = 3 1/3 XY = 5YZ = 20/3 = 6 2/3