Question

if in two triangles ABC and DEF , AB/DF =BC/FE=CA/ED, Then:
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Answer 1

Answer:

Δ ABC ~ Δ DFE

Step-by-step explanation:  

Given AB/DF =BC/FE=CA/ED

We have to find the names of the triangles which are similar

First, we shall name Δ ABC

Now, in Δ ABC, AB is corresponding with DF or FD

That leaves only one vertex for both cases, i.e C & E

∴ Δ ABC ~ Δ DFE  ....case I

or

Δ ABC ~ Δ FDE  ....case II

Lets take the ratio for both cases

Case I

By BPT, we get AB/DF =BC/FE=CA/ED

Case II

By BPT, we get AB/FD =BC/DE=CA/EF

As we are given the 3 ratios as AB/DF =BC/FE=CA/ED, case I is the correct answer.

∴ Δ ABC ~  Δ DFE

Answer 2

Given:

Two triangles ABC and DEF in which AB/DF =BC/FE=CA/ED.

To find:

The relationship between the given triangles ABC and DEF.

Solution:

As we know that according to the basic proportionality theorem (BPT), two triangles ABC and PQR are said to be similar if all the corresponding sides of the two triangles are in the same proportion. Thus, we have

$\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}$

Now, as given in the question, we have

$\frac{AB}{DF} = \frac{BC}{FE} = \frac{CA}{ED}$

By using BPT, we have

Side AB is corresponding to DF, BC is corresponding to FE and CA is corresponding to ED.

Hence, triangle ABC is similar to triangle DFE.