If OP is perpendicular to AB, prove that AP = BP. In a triangle ABC, D is the midpoint of BC. AD is produced up to E so that DE = AD. Prove that:
(i) Triangle ABD and triangle ECD are congruent.
(ii) AB = EC.
(iii) AB is parallel to EC.
Answer (1)
To solve this problem, we need to prove three parts involving a triangle ABC where D is the midpoint of BC, and AD is extended to a point E such that DE = AD.
Prove that △ A B D and △ EC D are congruent.
Since D is the midpoint of BC, we have B D = D C .
We are given that D E = A D .
Also, since AD is common to both triangles △ A B D and △ EC D , A D is equal in both triangles.
Therefore, by the Side-Side-Side (SSS) congruence postulate, △ A B D ≅ △ EC D .
Show that A B = EC .
Since △ A B D and △ EC D are congruent, all respective sides and angles are equal.
Thus, A B is equal to EC because corresponding parts of congruent triangles are equal (CPCTC).
Prove that AB is parallel to EC.
When two triangles are congruent, their corresponding angles are equal.
Therefore, ∠ A B D = ∠ EC D and ∠ A D B = ∠ C D E .
The equal angles ensure that AB and EC are corresponding sides of congruent angles, showing the sides are parallel because if a transversal intersects two lines such that alternate interior angles are equal, then the lines are parallel by the Alternate Interior Angles Theorem.
In conclusion, using the properties of congruent triangles and theorems about parallel lines, we have shown that △ A B D ≅ △ EC D , A B = EC , and AB is parallel to EC.
