Question

In the adjoining figure, AD and BC are equal perpendiculars to the line segment AB. Show that : CD bisects AB.

Answer 1

Given: AD and BC are perpendicular to AB and AD = BC.

To Prove: CD bisects AB (i.e., if CD meets AB at O then AO = OB).

Proof [Step-wise]:

1. Let O be the intersection point of CD and AB.

2. Since AD ⟂ AB and BC ⟂ AB, AD || BC both are perpendicular to the same line AB.

3. Lines AB and CD meet at O, so ∠AOD and ∠BOC are vertical (opposite) angles and therefore equal.

4. Because AD || BC and CD is a transversal, ∠ADO = ∠BCO (corresponding/alternate interior angles).

5. From steps 3 and 4, triangles AOD and BOC have two equal angles, so ΔAOD ∼ ΔBOC (AA similarity).

6. From similarity, corresponding sides are proportional:

`(AO)/(OB) = (AD)/(BC)`

7. Given AD = BC.

The ratio `(AD)/(BC) = 1`. 

Hence `(AO)/(OB) = 1`.

So AO = OB.

8. Therefore, O is the midpoint of AB, so CD bisects AB.

CD bisects AB proved.