State the basic proportionality theorem for 0
Answers
Answer:
Step-by-step explanation:
In the figure alongside, if we consider DE is parallel to BC, then according to the theorem,
ADBD=AECE
Let’s not stop at the statement, we need to find a proof that its true. So shall we begin?
PROOF OF BPT
Given: In ΔABC, DE is parallel to BC
Line DE intersects sides AB and AC in points D and E respectively.
To Prove: ADBD=AECE
Construction: Draw EF ⟂ AD and DG⟂ AE and join the segments BE and CD.
Proof:
Area of Triangle= ½ × base× height
In ΔADE and ΔBDE,
Ar(ADE)Ar(DBE)=12×AD×EF12×DB×EF=ADDB(1)
In ΔADE and ΔCDE,
Ar(ADE)Ar(ECD)=12×AE×DG12×EC×DG=AEEC(2)
Note that ΔDBE and ΔECD have a common base DE and lie between the same parallels DE and BC. Also, we know that triangles having the same base and lying between the same parallels are equal in area.
So, we can say that
Ar(ΔDBE)=Ar(ΔECD)
Therefore,
A(ΔADE)A(ΔBDE)=A(ΔADE)A(ΔCDE)
Therefore,
ADBD=AECE
Hence Proved.
The BPT also has a converse which states, if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
(Note: A converse of any theorem is just a reverse of the original theorem, just like we have active and passive voices in English.)
Read the properties of Triangles and Quadrilaterals here.
PROPERTIES OF BPT
The BPT has 2 properties.
Property of an angle bisector.
Property of Intercepts made by three parallel lines on a transversal.
Property of an Angle Bisector
Statement: In a triangle, the angle bisector divides the side opposite to the angle in the ratio of the remaining sides.
Angle bisector
In the given figure, seg AD is the angle bisector of ∠BAC.
According to the property,
BDDC=ABAC
Property of Intercepts made by three parallel lines on a transversal
Statement: The ratio of the intercepts made on the transversal by three parallel lines is equal to the ratio of the corresponding intercepts made on any other transversal of the same parallel line.
intercepts
Consider the above figure, line l, m, and n are parallel to each other. Transversals p and q intersect the lines at point A, B, C and D, E, F. So according to the property,
ABBC=DEEF
Learn more about Similarity of Triangles here.
Solved Example
Q: In triangle ABC, seg AD is the angle bisector of∠BAC. BD=6, DC=8, AB=15 Find AC
triangle
Solution:
segments AD bisects ∠BAC (given)
.’. AB/AC=BD/DC (Angle bisector property)
Assume AC= x
.’. 15/x=6/8
.’. 15 × 8 =6 × x
.’. x= (15 × 8) / 6 = 20.
Therefore, AC= 20.
Q: We are given that in triangle PQR, MN intersects PQ and PR at M and N respectively such that PM = 3 cm, MQ = 9 cm, PN = 2 cm and NR = 6 cm. Is MN parallel to QR?
Solution:
PM= 3 cm, MQ=9 cm
Now, PM/MQ=3/9=1/3
Also, PN=2 cm, NR=6 cm.
Now, PN/NR= 2/6=1/3
Therefore, PM/MQ=PN/NR
Hence, by the converse of basic proportionality theorem, we have MN parallel to QR.
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Congruent TrianglesInequalities of TriangleProperties of TrianglesSimilarity of TrianglesBasic Proportionality Theorem and Equal Intercept TheoremPythagoras Theorem and Its Applications
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Triangles
Congruent Triangles
Inequalities of Triangle
Properties of Triangles
Similarity of Triangles
Basic Proportionality Theorem and Equal Intercept Theorem
Pythagoras Theorem and Its Applications
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