CBSE:10th Math :Chapter Triangles : Solved Concept builder theorems
If a line divides any two sides of a triangles proportionally then the line is parallel to the third side
Statement: If a line divides any two sides of a triangles proportionally (in same ratio), then the line is parallel to the third side.
Proof: We are given ABC
Proof: We are given ABC
AD / BD = AE / CE
We have to prove: DE is parallel to BC
Let DE's not parallel to BC then an another line DE' is parallel to BC.
Now
AD / BD = AE / CE [Given]
And AD / BD = AE' / CE' [Thales Theorem]
Therefore AE / CE = AE' / CE'
i.e. AE / CE + 1 = AE' / CE' + 1
i.e. AE + CE / CE = AE' + CE' / CE'
i.e. CE = CE'
But this is not possible until E and E' is coincident.
Thus, our assumption is wrong and DE is parallel to BC.
We have to prove: DE is parallel to BC
Let DE's not parallel to BC then an another line DE' is parallel to BC.
Now
AD / BD = AE / CE [Given]
And AD / BD = AE' / CE' [Thales Theorem]
Therefore AE / CE = AE' / CE'
i.e. AE / CE + 1 = AE' / CE' + 1
i.e. AE + CE / CE = AE' + CE' / CE'
i.e. CE = CE'
But this is not possible until E and E' is coincident.
Thus, our assumption is wrong and DE is parallel to BC.
X Triangles :Criteria for Similarity of Triangles Solved:
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