In ABC- DE - BC and CD - EF- Prove that AD2 - AF - AB-

We have to prove that AD^2=AF × AB Consider ABC, DE∥ BC by basic proportionality theorem. #160;If a line is parallel to a side of a triangle which ...

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Standard X

Mathematics

Basic Proportionality Theorem

In ABC, DE ...

Question

In $△ A B C, D E ∥ B C$ and $C D ∥ E F$ . Prove that $A D^{2} = A F \times A B$ .

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Solution

We have to prove that $A D^{2} = A F \times A B$
Consider $△ A B C, D E ∥ B C$ by basic proportionality theorem.
If a line is parallel to a side of a triangle which intersects the other sides into two distinct points, then the line divides the sides in proportion.
i.e, $\frac{A D}{D B} = \frac{A E}{E C}$ .....(i)
From $△ A D C, E F | | D C$
$∴,$ By Basic Proportionality Theorem, $\frac{A F}{F D} = \frac{A E}{E C}$ .....(ii)
From (i) and (ii), $\frac{A D}{D B} = \frac{A F}{F D}$
$\Rightarrow \frac{D B}{A D} = \frac{F D}{A F}$ [Taking Reciprocal]....(iii)
Adding 1 on both sides of equation (iii),
$\frac{D B + A D}{A D} = \frac{F D + A F}{A F}$
$\frac{A B}{A D} = \frac{A D}{A F}$
$∴ (A D)^{2} = A F \times A B$

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In △ABC,DE∥BC and CD∥EF. Prove that AD2=AF×AB.

In $△ A B C, D E ∥ B C$ and $C D ∥ E F$ . Prove that $A D^{2} = A F \times A B$ .