Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).

Theorem 6.2: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

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Solution

Given: In $△ A B C, D$ and $E$ are midpoints of $A B$ and $A C$ respectively,
i.e., $A D = D B$ and $A E = E C$
To Prove: $D E ∥ B C$
Proof:
Since, $A D = D B$
$∴$ $\frac{A D}{D B} = 1............ (1)$
Also,
$A E = E C$
$∴$ $\frac{A E}{E C} = 1............ (2)$
From $(1)$ and $(2)$ ,
$\frac{A D}{D B}$ $=$ $\frac{A E}{E C}$ $= 1$
i.e., $\frac{A D}{D B}$ $=$ $\frac{A E}{E C}$
$∴$ By converse of Basic Proportionality theorem,
$D E ∥ B C$

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Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).Theorem 6.2: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

Given: In △ABC,D and E are midpoints of AB and AC respectively,i.e., AD=DB and AE=ECTo Prove: DE∥BCProof: Since, AD=DB∴ADDB=1............(1)Also,AE=EC∴AEEC=1............(2)From (1) and (2),ADDB = AEEC =1i.e.,ADDB = AEEC∴ By converse of Basic Proportionality theorem,DE∥BC

Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).

Theorem 6.2: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.