Question

In the given figure, \[AD = AE\] and $A{D^2} = BD \times EC$. Prove that triangles ABD and CAE are similar.

Answer 1

Hint:Use the rule of the triangle that the corresponding angles of the equal sides are also equal using the given data \[AD = AE\] and $A{D^2} = BD \times EC$ to get the required result to show the congruency of the triangle.

Complete step-by-step answer:
It is given in the problem that \[AD = AE\] and $A{D^2} = BD \times EC$ in the figure and we have to show that triangles ABD and CAE are similar triangles.

We know that two triangles are similar if they follow any one of the congruence criteria; AAA, SSS, and SAS.
It is given in the adjoint figure that \[AD = AE\], then the corresponding angles are also the same in the triangle $ADE$. That is,
$\angle ADE = \angle AED$
Then we can also conclude in the triangles $ABD$ and $ACE$:
$\angle ADB = \angle AEC$ (Linear pair angles)
It is also given to us in the problem that:
$A{D^2} = BD \times EC$
We can also write it as:
$\dfrac{{AD}}{{BD}} = \dfrac{{EC}}{{AD}}$
Substitute the given value \[AD = AE\], then we have
$\dfrac{{AD}}{{BD}} = \dfrac{{EC}}{{AE}}$
The above relation shows that the two sides of the triangles $ABD$ and $ACE$ are in the same proportion.
Now, in the triangles $ABD$ and $ACE$, we have
$\angle ADB = \angle AEC$
$\dfrac{{AD}}{{BD}} = \dfrac{{EC}}{{AE}}$
Then using the SAS (Side-Angle-Side) criteria of congruence, we got the result that the triangles $ABD$ and $ACE$ are similar to each other.
This is the required result.

Note:Any two triangles are similar then they have a similar shape it is not mandatory that they also have the same size, the size may vary. There are three criteria to show that the given triangles are similar-
(i) AAA: angle-angle-angle
(ii) SSS: side-side-side
(iii) SAS: side-angle-side
We can use three criteria to show that the given triangles are similar.