Question

In a triangle ABC, D is the midpoint of BC; AD is produced up to E so that DE= AD.
Then AB=EC
A.True
B.False

Answer 1

Hint: We would start by drawing a figure with the help of the given information. Then, to know whether AB = EC, we would try to prove the congruence of two triangles having AB and EC as one of their sides respectively. If those triangles are proved to be congruent then by CPCTC rule, these two sides would definitely be equal , else, if not congruent then, the answer would become ‘false’.

Complete step-by-step answer:
Given: A triangle ABC in which, D is the midpoint of BC. AD is produced up to E so that DE= AD.
To find: Whether AB is equal to EC or not.
First of all, we will draw a triangle ABC such that D is the midpoint of BC
\[ \Rightarrow \] BD = CD (because midpoint divides a line into two equal parts)
Now, we will produce AD up to E so that, DE = AD.
Also, we will join EC to complete the triangle ECD which would help us to find our required answer.
Now, to show whether AB is equal to EC or not,
We will try to prove that $\vartriangle ABD \cong \vartriangle ECD$

Now, in $\vartriangle ABD$ and $\vartriangle ECD$ ,
BD = CD (given)
AD = ED (given)
$\angle ADB = \angle EDC$ (vertically opposite angles)
Hence, $\vartriangle ABD \cong \vartriangle ECD$by Side-Angle-Side (SAS) congruence rule.
Because, if the length of two pairs of sides of two triangles are equal, and the included angles are also equal, then the triangles are congruent by SAS rule.
Also, if two triangles are congruent, then the corresponding parts of congruent triangles ( also known as CPCTC) are also equal.
Therefore, by CPCTC,
AB = EC
(Since, $\vartriangle ABD \cong \vartriangle ECD$ , therefore their corresponding sides and angles will be equal.)
Hence, the given statement is true.
Therefore, option A is the correct answer.

Note: In these types of questions, it is mandatory to write all the given information and what is required to be proved. Also, drawing the figure while reading the given information, helps us to make a better and clearer image of what we are required to prove. This question cannot be solved without knowing the concepts like: how to apply congruence rules, how to show that two triangles are congruent, SSS, ASA, AAS and SAS rules.
Also, after proving that two triangles are congruent, we should know that their corresponding sides and angles will be equal by the CPCTC theorem.