To determine the shape of quadrilateral ABCD, we must verify the properties by calculating distances and slopes. The vertices are A(0, 0), B(4, 4), C(4, 8), and D(0, 4).
Step 1: Calculate Side Lengths:
- AB = √((4-0)² + (4-0)²) = √(16 + 16) = √32 = 4√2
- BC = √((4-4)² + (8-4)²) = √(0 + 16) = √16 = 4
- CD = √((4-0)² + (8-4)²) = √(16 + 16) = √32 = 4√2
- DA = √((0-0)² + (4-0)²) = √(0 + 16) = √16 = 4
Step 2: Check for Right Angles Using Slopes:
- Slope of AB = (4-0)/(4-0) = 1
- Slope of BC = (8-4)/(4-4) is undefined, indicating a vertical line.
- Slope of CD = (8-4)/(4-0) = 1
- Slope of DA = (4-0)/(0-0) is undefined, indicating a vertical line.
Conclusion: Opposite sides AB & CD and BC & DA are equal, and perpendicular pairs (since slopes of perpendicular lines are negative reciprocals or one is vertical and the other is horizontal). Thus, ABCD is a rectangle.
Important Questions from Miscellaneous (APSC)
- On November 21, 2024, the Assam Government officially renamed the Karimganj district as Sribhumi, along with its headquarters town Karimganj, which will now be called Sribhumi town. The new name was inspired by Rabindranath Tagore's description of the area over a century ago as 'Sribhumi'. As per Kabiguru Tagore's description, the word 'Sribhumi' is referred to as
- The Assam Government's decision to officially recognize 'Bathouism' in the Bodoland Territorial Region (BTR) primarily aims to
On February 24, 2025, Prime Minister Narendra Modi attended the largest Jhumur event in history, celebrating the 200th anniversary of Assam's tea industry. Consider the following statements :
(i) The tea garden community migrated from Central India in the 19th century to work in the tea gardens of Assam.
(ii) Jhumur dance originates from the Sadan ethnolinguistic group of the Chota Nagpur region.
(iii) The songs sung during Jhumur performances often depict the struggles of tea plantation workers. They narrate stories of migration and exploitation, depicting the community's socio-economic challenges.
(iv) Jhumur dance plays a vital role in tea garden festivals, particularly during Tushu Puja and Karam Puja, which celebrate the harvest time. Which of the above statements is/ are not correct?
- Which of the following statements about Kaziranga National Park is incorrect?
- In February 2025, a significant conservation milestone was achieved in Assam concerning the Greater Adjutant Stork. Which of the following best describes this achievement?
