In the following question, a sentence is given in direct speech. Select the option that best expresses the same sentence in indirect (reported) speech. My father said, "We have been living here since the day this house was built."

Introduction / Context:
This sentence reports a long lasting action that started in the past and continues up to the present of the original speaker. It uses present perfect continuous and a past passive clause about the house being built. In indirect speech with a past reporting verb, both parts must be backshifted and place and demonstrative words must be adjusted. The question checks understanding of these advanced aspects of reported speech.

Given Data / Assumptions:

• Reporting verb: "said" in simple past.
• Direct speech: "We have been living here since the day this house was built."
• Pronoun "we" refers to "they" in reported speech.
• Verb phrase "have been living" is present perfect continuous.
• Clause "this house was built" is a simple past passive.
• Place word "here" and demonstrative "this house" must change appropriately.

Concept / Approach:
When the reporting verb is in the past, present perfect continuous "have been living" usually changes to past perfect continuous "had been living". The passive clause "was built" is often backshifted to "had been built". The place word "here" changes to "there" and "this house" becomes "that house" to match the different point of view. The prepositional phrase "since the day" remains, as it correctly shows the starting point of the action.

Step-by-Step Solution:
Step 1: Start with "My father said that" for the reporting clause. Step 2: Change "we have been living" to "they had been living", which backshifts the tense and changes the pronoun. Step 3: Replace "here" with "there" to indicate that the place is being talked about from a different location or time perspective. Step 4: Change "since the day this house was built" to "since the day that house had been built". "This house" becomes "that house" and "was built" becomes "had been built". Step 5: The full reported sentence is "My father said that they had been living there since the day that house had been built."

Verification / Alternative check:
The reported sentence clearly indicates that the family started living in the same place when the house was built and had continued to do so up to the time of the original statement. By using past perfect continuous, we show that this ongoing action started before the time of reporting. The passive "had been built" correctly places the building of the house even earlier. Changes from "here" to "there" and "this" to "that" indicate that the description is no longer spoken at the original location.

Why Other Options Are Wrong:
Option A keeps "here" and does not backshift the passive clause, so it is incomplete in its changes. Option C uses "have been living" and "were built", both of which do not follow the needed backshifts and contain a subject verb agreement error in "house were built". Option D again keeps present perfect continuous and uses "were being built", which changes the meaning and makes the building action look incomplete or ongoing, which is not true in the original sentence.

Common Pitfalls:
Many learners find it difficult to handle present perfect continuous in reported speech, especially when it is paired with a "since" clause. Another typical mistake is to forget to change "here" and "this" in such sentences because the focus tends to remain on tense changes. Passive structures also cause confusion, leading some learners to drop "been" or to mishandle agreement. Careful attention to each element helps ensure the reported sentence keeps both correct grammar and original meaning.

Final Answer:
The correct reported sentence is My father said that they had been living there since the day that house had been built.