Similar Triangles in Grade VI

Hello friends, in today's session we are going to learn about the similar triangles and how to solve problems based on proportional reasoning (some portion taken from ICSE class 9 syllabus).
Similar triangles are those triangles which have same shape but different size. If the size of the triangles also become equal then they are no more called as similar but are called as Congruent triangles.
The same definition is true even for other Polygons. The similarity between two figures is represented by using a symbol ‘<’, which says “ is similar to”.
Just like to find congruency we use different methods in a same manner to find similarity we have different methods which are very much similar to the congruency.

1. AAA( Angle Angle Angle) – If all the three angles of one triangle are equal to the three angles of the other triangle then they can be called similar.
   
2. SSS( Side Side Side) – If all the three sides of one triangle are equal to the three sides of the other triangle, then the triangles are said to be similar.
   
3. SAS( Side Angle Side) – If two sides and the included angle of one triangle are in the same ratio with the other triangle then also they are similar.
   

If we take two triangles ∆ABC and ∆DEF, then we can find that they are similar or not using the above three properties. If they are similar then they can be written as ∆ABC < ∆DEF.
Now how to use proportional reasoning in similarity and solve problems.
Let's understand this with the help of an example.
Which score among this is better?
50 runs on 10 balls or 40 from 10 balls.
In this case we can easily say that 50 runs from 10 balls is better. As the ratio for the first one is 50/10 = 5 and the ratio for the second one is 40/10 = 4. So 5 is greater than 4 so the first one is better.
For solving problems based on proportional reasoning we have to follow some simple steps.

1. describe the given ratio in words,
   
2. convert them into same unit of measurement.
   
3. put them in the ratio form, and write the missing one as x.
   
4. find the value of x by cross multiplication or by any other method.
   
5. Convert in the appropriate unit of measurement.
   

For Example -

1. ∆ EFG is similar to ∆ XYZ. The sides of ∆ EFG measure 5, 6, and 14. Two sides of ∆ XYZ measure 15 and 18. The third side measures
   

Now to solve this problem we will use proportional reasoning.
So will be 5/15 = 6/18 = 14/x.
1/3 = 14/x.
x = 14 x 3.
x = 42.
So this is the length of the third side of the triangle.

2. if ∆ ABC is similar to ∆ MNO. AB/MN = BC/NO =?
   

As the all the sides of the triangle are in a similar ratio. So the ratio will be AB/MN = BC/NO = AC/MO.
So our answer is AC/MO.

In upcoming posts we will discuss about Math Blog on Grade VI. Visit our website for information on rational expressions