Introduction of basic proportionality theorem:
The Basic proportionality theorem is the theorem associated to triangle's property. This theorem says that if a line intersects two sides of one triangle and is parallel to the third side, then it divides the first two sides proportionally. That’s the Basic Proportionality Theorem shows the relation between the triangle and a straight line which bisects the two sides and parallel to another side.
Explanatory of basic proportionality theorem:
If a line is drawn parallel to the triangle, thus the line intersect two sides of the triangle parallel to the other (third) side in distinct points, then it cuts(divide) the two sides proportionally in the same ratio. This is known as proportionality theorem. It is otherwise called as Thales theorem.
Now we are going to prove the theorem and illustrative the statements
Given: A triangle LMN, S and T are any other two points on PQ and PR respectively such that ST parallel to QR.
To prove:
LS / SM = PT / TN
Construction:
Join MT and NS. From TU perpendicular to LM.
Proof:
In triangle LMN
ST is parallel to MN
And
If LS/MS = k (Where k is constant)
Thus LT / NT is also equal to k
Or we can say
LS / MS = LT / NT
We are given LMN,
ST is parallel to MN.
Understanding sum of the squares formula is always challenging for me but thanks to all math help websites to help me out.
Basic proportionality theorem:
Proportionality Theorem says that, A line parallel to one side of a triangle divides the other two sides into parts of equal proportion.
DATA:
In triangle ABC, a line PQ drawn parallel to BC cuts AB and AC at P and Q respectively.
To prove:
AP/ PB = AQ /QC
Construction:
Let the point P divide AB in the ratio of l: m where l and m are natural numbers. Divide AP into 'l' and PB into 'm' equal parts. Through all these points on AB, draw lines parallel to BC to cut AC.
Proof:
Statement Reason
1. AP is cut in l equal parts and lines are drawn through Construction
these points parallel to BC.
2. Therefore, AQ is cut l equal parts. Intercept theorem
3. Similarly, Statement 1
QC is cut in m equal parts.
4. AP / PB = 1 /m Construction
5. AQ / QC = 1/ m Statement 2 and 3
6. Therefore AP / PB = AQ / QC Statement 4 and 5.
The Basic proportionality theorem is the theorem associated to triangle's property. This theorem says that if a line intersects two sides of one triangle and is parallel to the third side, then it divides the first two sides proportionally. That’s the Basic Proportionality Theorem shows the relation between the triangle and a straight line which bisects the two sides and parallel to another side.
Explanatory of basic proportionality theorem:
If a line is drawn parallel to the triangle, thus the line intersect two sides of the triangle parallel to the other (third) side in distinct points, then it cuts(divide) the two sides proportionally in the same ratio. This is known as proportionality theorem. It is otherwise called as Thales theorem.
Now we are going to prove the theorem and illustrative the statements
Given: A triangle LMN, S and T are any other two points on PQ and PR respectively such that ST parallel to QR.
To prove:
LS / SM = PT / TN
Construction:
Join MT and NS. From TU perpendicular to LM.
Proof:
In triangle LMN
ST is parallel to MN
And
If LS/MS = k (Where k is constant)
Thus LT / NT is also equal to k
Or we can say
LS / MS = LT / NT
We are given LMN,
ST is parallel to MN.
Understanding sum of the squares formula is always challenging for me but thanks to all math help websites to help me out.
Basic proportionality theorem:
Proportionality Theorem says that, A line parallel to one side of a triangle divides the other two sides into parts of equal proportion.
DATA:
In triangle ABC, a line PQ drawn parallel to BC cuts AB and AC at P and Q respectively.
To prove:
AP/ PB = AQ /QC
Construction:
Let the point P divide AB in the ratio of l: m where l and m are natural numbers. Divide AP into 'l' and PB into 'm' equal parts. Through all these points on AB, draw lines parallel to BC to cut AC.
Proof:
Statement Reason
1. AP is cut in l equal parts and lines are drawn through Construction
these points parallel to BC.
2. Therefore, AQ is cut l equal parts. Intercept theorem
3. Similarly, Statement 1
QC is cut in m equal parts.
4. AP / PB = 1 /m Construction
5. AQ / QC = 1/ m Statement 2 and 3
6. Therefore AP / PB = AQ / QC Statement 4 and 5.
