Question

If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third

side.

1. In triangle (ABC) , if D is midpoint of AB and DE// BC , then find the ratio of AE : ED.

2. Draw the rough diagram of given data.​

Answer 1

Given:

In ΔABC, D is a midpoint of AB

DE // BC

To find:

1. The ratio of AE : ED

2. Draw the rough diagram of given data.

Solution:

(1). Finding the ratio of AE : ED :-

We know that,

$\boxed{\bold{Converse\: of\:Basic\:Proportionality\: Theorem}}$ : If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

∴ ....... (i)

Since D is a midpoint of AB (given)

∴ AD = DB ...... (ii)

From (i) & (ii), we get

AE = EC

⇒ E is a midpoint of AC ..... (iii)

Now,

Consider Δ AED and ΔACB, we have

∠A = ∠A ........ [common angle]

∠ADE = ∠ABC ..... [corresponding angles, ∵DE//BC and side AB forms a transversal]

∴ Δ AED ~ ΔACB ........ By AA Similarity

Also, we know that the corresponding sides of two similar triangles are proportional to each other.

∴  $\frac{AE}{AC} = \frac{ED}{BC}$  

on rearranging, we get

⇒ $\frac{AE}{ED} = \frac{AC}{BC}$

⇒ $\frac{AE}{ED} = \frac{AE + EC}{BC}$

⇒  $\boxed{\bold{AE : ED = [AE + EC] : BC}}$

(2). Diagram of the given data:-

Rough diagram is attached below

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