Hello math bugs(🐞) & hivers(🐝)

Well come to another episode of geometry. Today, I have come up with a problem on similarity. It is very easy of you did this before. 

At first, let me elaborate the question a little bit. Here is random trinagle called ∆ABC. D and E are two points on sides AB and AC respectively drawn such a way that ***DE is Parallel to BC***. D is also a point on AC such that AF becomes 2.5 cm and FD becomes 3.5cm. Here ***EF is also parallel to BD*** as you can see in the problem figure given below. Now, You have to find the value of **AC**

![](https://images.ecency.com/DQmWv8nTe8h7h3kQhGD9ghhCsvQXwMRuCbxXALrJraTwKZS/addtext_12_05_09.52.08.jpg)

The only thing you need to know here is Thales theorem. So, let me explain it first, then I'll go to soluion. 

***Thales theorem***:

The basic concept is if a parallel line of any side of a trinagle is drawn, the line cut the other two sides in equal ratio. It is also called basic proportionality theorem. Check the following figure 👇

![](https://images.ecency.com/DQmNQy3qoYqNWTZtsEQ32jZJhqqWcSSnyXmnkDAgeipcphq/addtext_12_06_12.01.03.png)

***It can be proved using similarity. Let's do it***

In the above figure, you can see two triangle namely ∆ABC and ∆AED. If we compare angles of them , we can see all of the respective anagles are equal. 

∠BAC = EAD [ common angles]
∠ABC = ∠AED [ similiar angles as BC||ED and AB transversal]
∠ACB = ∠ADE [ similiar angles as BC||ED and AC transversal]

Hence,  ∆ABC ~ ∆AED 

So , the ratios of correspoding sides are equal.

**Note**:

✍️ Reverse of thales theorem is also ture: **If AE/EB= AD/DC then ED || BC**    

✍️We not only get AE/EB =AD/DC, we can have other ratios: **AB/AE = AC/AD or AB/EB =AC/DC**    

✍️If the points E and D would be mid points of AB and AC respectively, **the ratios would 1:1 and in that case ED would be half of BC**.


**SOLUTION**:

First, fonsider ∆ABD. In ∆ ABD EF|| BD so AF : FD = AE : EB or **AF : AD = AE : AB**. As we need to find AC , then 2nd one will work well. So we can say as follows:


![](https://images.ecency.com/DQmRpQYmLsdUPGTBitCNuoN3FRc1ixQm9HzueBcc88VPpwa/addtext_12_07_01.24.46.png)

          AE : AB = AF : AD

      Or, AE : AB = AF : (AF + FD)

      Or, AE : AB = 2.5 : (2.5 + 3.5)

      Or, AE : AB = 2.5 : 6
   
      Or, AE : AB = 5 : 12


Let AE = 5y cm and AB 12y cm

Now, in ∆ ABC ED || BC , so again we can say AE : AB = AD : AC. Check it below👇

![](https://images.ecency.com/DQmaBQesj1UTDNvwUC1HYWkWjYRarRsWGq2mHf7WfXKSCf3/addtext_12_07_01.34.21.png)

          AE : AB = AD : AC 

      Or, 5y : 12y = 6 : AC 

      Or, AC = 12y × 6/ 5y

      Or, AC = 72/5
  
      Or, AC = 14.4  [ cm]


If you understand stand thales theorem or similarity, the problem can be done in seconds. 

**Short cut approach**:

     Direct from problem figure

      5 units = 6 cm
     
      1 unit = 6/5 cm

      12 units = 6/5× 12 cm

     So, AC = 14.4 cm


🎤🎤***Figures used here may mot be accurate, try considering just the value given. All the figures are given here are made by only***.

I hope you liked my explanation 

Thank you so much for visitng 

Have a great day 

All is well

Regards: @meta007