Question

In a triangle ABC with\[\angle A\angle B\angle C\], points D, E, F are on the interior of segments BC, CA, AB respectively. Which of the following triangles cannot be similar to ABC?

A) Triangle ABD
B) Triangle BCE
C) Triangle CAF
D) Triangle DEF

Answer 1

Hint:
Here, we will find the triangle which is not similar to triangle ABC. We will check the similarity of the triangles by checking whether their three sides are in proportion or not. We will write the three sides of both triangles in ratios and check whether they satisfy the same proportion rule. If they don’t satisfy that means they are not similar triangles and we will get our answer.

Complete step by step solution:
First, we will check the option (A) as follows,


For similarity of triangles \[\Delta ABC \sim \Delta ABD\], the ratio of the corresponding sides should be equal. Therefore, we get
\[\dfrac{{AB}}{{AB}} = \dfrac{{BC}}{{BD}} = \dfrac{{AC}}{{AD}}\]
\[ \Rightarrow \dfrac{1}{1} = \dfrac{{BC}}{{BD}} = \dfrac{{AC}}{{AD}}\]
From the above ratio \[BC = BD\] but that is not possible as D lies in between BC and \[BC > BD\].
So, \[\Delta ABD\] is not similar to \[\Delta ABC\].

Hence, option (A) is correct.

Note:
Similar triangles are the triangles having the same shape but different sizes. For two triangles to be similar their corresponding angles should be congruent and corresponding sides should be in proportion. The symbol used to denote similarity of triangles is \[ \sim \]. There are many ways to determine whether two triangles are similar or not which are defined as follows:
1) If two angles of one triangle are equal to the other triangle they are similar.
2) If two sides are in the same proportion in both triangles and the angle formed by the two sides are also the same then they are similar triangles.
3) If all three sides of the two triangles are in proportion they are similar triangles.