Name: Example on Thales Theorem
Uploaded: 2019-07-19
Duration: 4 min 26 s
Description: If the midpoint of two sides of a triangle is joined, then this line segment is parallel to the third side. Let's explore more by an example in this video.

Question

Prove that if a line is drawn parallel to one side of triangle to intersect the other two points in distinct point the other two points are divided in the same ratio.

Accepted Answer

Suppose a $△ A B C$ in which a line $D E$ parallel to $B C$ intersects $A B$ at $D$ and $A C$ at $E$ .

To prove: $D E$ divides the two sides in the same ratio.

$\frac{A D}{D B} = \frac{A E}{E C}$

Construction: Join BE, CD and draw $E F ⊥ A B$ and $D G ⊥ A C$ .
Proof: Here, $\frac{ar (△ ADE)}{ar (△ BDE)} = \frac{\frac{1}{2} \times AD \times EF}{\frac{1}{2} \times DB \times EF}$ $[∵ area of triangle = \frac{1}{2} \times base \times height]$

$\Rightarrow \frac{ar (△ ADE)}{ar (△ BDE)} = \frac{A D}{D B} . . . (i)$

Similarly, $\frac{a r (△ A D E)}{a r (△ D E C)} = \frac{\frac{1}{2} \times A E \times G D}{\frac{1}{2} \times E C \times G D} = \frac{Α E}{E C} . .. (i i)$

Now, since, $△ B D E$ and $△ D E C$ lie between the same parallel $D E$ and $B C$ and on the same base $D E$

So, $a r (△ B D E) = a r (△ D E C) . .. (i i i)$

From Equations $(i), (i i)$ and $(i i i)$ , we get,

$\frac{A D}{D B} = \frac{A E}{E C}$

Hence, proved.

Prove that if a line is drawn parallel to one side of triangle to intersect the other two points in distinct point the other two points are divided in the same ratio.

Important Questions on Triangles

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