Triangle Proportionality Theorem
Triangle proportionality theorem,Implementation of the Theorem
The triangle proportionality theorem states that, when a straight line is drawn inside the triangle tin such a way it is parallel to any one side of the triangle, it proportionally divides the other two sides. The line drawn may be parallel to any of the sides. The concept of triangle proportionality theorem is used when some variables are unknown, since the line divides the other two sides proportionally. Proof is nothing but the condition that satisfies properties of the given figure.
Triangle Proportionality Theorem:
Consider a triangle ABC, where BC is the base of the triangle. If a line DE is drawn parallel to the base (BC), therefore DE || BC.
TRIANGLE PROPORTIONALITY THEOREM
Then according to the triangle proportionality theorem
' (AE)/(AC)' = '(AD)/(AB)'
Implementation of the Theorem:
Ex:1 Consider a triangle ABC, where BC is the base of the triangle. If a line DE is drawn parallel to the base (BC), therefore DE || BC.
If AD = 3 cm, EC = 4 cm, DB = 11 cm, and BC = 12 cm, then find the length of AP rounded to the nearest decimal.
Sol:
Triangle 1
' (AE)/(AC)''(AD)/(AB)'[Proportionality Theorem]
'[((AE)/(AE+EC))]' = '[((AD)/(AD+DB))]' [From the figure.]]
Substituting the values,
'[((AE)/(AE+4))]''[((3)/(3+11))]'
On Cross multiplying, we get,
3(AE) + 11(AE) = 3(AE) + 12,
On simplifying we get,
11(AE) =12
Dividing by 11 on both sides we get,
'(11(AE))/11' ='(12)/11'
AE =1.09
Ex:2 Consider a triangle PQR, where QR is the base of the triangle. If a line ST is drawn parallel to the base (QR), therefore DE || BC. The values are PS =4, PT =4, TR =!2, Find the value of SQ.
Sol:
Triangle 2
' (PS)/(PQ)''(PT)/(PR)'[Proportionality Theorem]
'[((PS)/(PS+SQ))]' = '[((PT)/(PT+PR))]' [From the figure.]]
Substituting the values,
'[((4)/(SQ+4))]''[((4)/(4+12))]'
On Cross multiplying, we get,
16 + 48 = 4(SQ) +16,
64 = 4(SQ) +16,
4(SQ) = 64-16
4(SQ) = 48
Dividing by 4 on Both sides,
'(4(SQ))/4' = ' 48/4'
On simplifying we get,
SQ = 12
Hence the proportionality theorem is explained and implemented.
Triangle Proportionality Theorem
Definition of Triangle Proportionality Theorem
Triangle Proportionality Theorem states that a line drawn parallel to any of the sides of a triangle divides the other two sides proportionally.
Example of Triangle Proportionality Theorem
In the given triangle ABC, BC is the base of the triangle.
DE is drawn parallel to BC and it intersects the other two sides AB and AC at D and E respectively. Here, = .
This is called triangle proportionality theorem.
by: Smith
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