Transcript triangle proportionality theorem

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© Boardworks 2012
Triangle proportionality theorem
Triangle proportionality theorem and converse:
A line is parallel to the side of a triangle and intersects the two
other sides if and only if it divides the sides proportionally.
State the triangle
proportionality theorem.
A
If XY is parallel to BC...
X
Y
... then AX/XB = AY/YC.
State the converse of the triangle
proportionality theorem.
B
C
If AX/XB = AY/YC...
... then XY is parallel to BC.
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Triangle proportionality theorem proof
Triangle proportionality theorem:
If a line is parallel to the side of a triangle and intersects the
two other sides, then it divides the sides proportionally.
Prove the triangle proportionality theorem.
given: XY ∥ BC
A
X
corresponding
∠AXY ≅ ∠B, ∠AYX ≅ ∠C
angles postulate:
AA similarity △ABC ~ △AXY
postulate:
⇒ AX/(AX+XB) = AY/(AY+YC)
Y
cross-multiply: AX·AY+AX·YC = AY·AX+AY·XB
B
C
simplify: AX·YC = AY·XB
divide by YC·XB: AX/XB = AY/YC
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
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Converse of the triangle prop. theorem
Converse of the triangle proportionality theorem:
If a line divides two sides of a triangle proportionally, then it
is parallel to the other side.
Prove the converse of the triangle proportionality theorem.
given:
cross-multiply :
A
add AX·AY:
X
Y
factor:
substitute:
rearrange:
reflex. prop.:
C SAS similarity
postulate:
B
conv. corr. ang. theorem:
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AX/XB = AY/YC
AX·YC = AY·XB
AX·YC+AX·AY = AY·XB+AX·AY
AX(AY+YC) = AY(AX+XB)
AX(AC) = AY(AB)
AX/AB = AY/AC
∠A ≅ ∠ A
△ABC ~ △AXY
⇒ ∠AXY ≅ ∠B, ∠AYX ≅ ∠C
XY ∥ BC 
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Summary problem
In the triangle shown, find the values of x and y.
x+4
Since DE∥FG and BC∥FG, use the
triangle proportionality theorem.
2x
y
D
F
A
9
E
12
G
7
C
B
First look at △AFG.
Then look at △ABC.
by the tri.
by the tri.
AD/DF = AE/EG prop. theorem: AF/FB = AG/GC
prop. theorem:
substitute
substitute (x + 4)/2x = 9 /12
known lengths (x + 4 + 2x)/y = (9 + 12)/7
known lengths: 12x + 48 = 18x
and x : 28/y = 21/7
solving for x: x = 8
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solving for x: y = 28 × 7 ÷ 21 = 8.05
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