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D and E are the mid-points of the sides AB and AC of △ABC and O is any point on the side BC, O is joined to A. If P and Q are the mid-points of OB and OC res, Then DEQP is
- a)A Rectangle
- b)A Triangle
- c)A Rhombus
- d)A Parallelogram
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
D and E are the mid-points of the sides AB and AC ofABC and O is any p...
In ΔABC, D and E are the mid-points of sides AB and AC, respectively.
By mid-point theorem,
DE || BC .....(i)
⇒ DE = PO + OQ
⇒ DE = PO + OQ
⇒ DE = PQ
Now, In ΔAOC, Q and E are the midpoints of OC and AC respectively.
From equations (iii) and (iv),
Now, In ΔAOC, Q and E are the midpoints of OC and AC respectively.
From equations (iii) and (iv),
EQ || PD and EQ = PD
From equations (i) and (ii),
DE || BC (or DE || PQ) and DE = PQ
Hence, DEQP is a parallelogram.
Most Upvoted Answer
D and E are the mid-points of the sides AB and AC ofABC and O is any p...
To prove that DEQP is a parallelogram, we need to show that both pairs of opposite sides are parallel.
1. Proving DE || QP:
Since D and E are the midpoints of sides AB and AC, respectively, we can use the Midpoint Theorem to show that DE is parallel to BC. According to the Midpoint Theorem, the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.
Therefore, DE || BC.
Now, let's consider triangle AOC. Since P is the midpoint of OB and Q is the midpoint of OC, we can apply the Midpoint Theorem again to show that PQ is parallel to AC.
Therefore, PQ || AC.
2. Proving DE || PQ:
To show that DE is parallel to PQ, we need to prove that BC is parallel to AC.
Consider triangle AOB. Since D is the midpoint of AB, we can use the Midpoint Theorem to show that DE is parallel to OA (extended). Therefore, DE is parallel to BC.
Now, let's consider triangle AOC. Since Q is the midpoint of OC, we can apply the Midpoint Theorem to show that PQ is parallel to OA (extended). Therefore, PQ is parallel to AC.
Since both pairs of opposite sides are parallel (DE || BC and PQ || AC), we can conclude that DEQP is a parallelogram.
Therefore, the correct answer is option D, a parallelogram.
1. Proving DE || QP:
Since D and E are the midpoints of sides AB and AC, respectively, we can use the Midpoint Theorem to show that DE is parallel to BC. According to the Midpoint Theorem, the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.
Therefore, DE || BC.
Now, let's consider triangle AOC. Since P is the midpoint of OB and Q is the midpoint of OC, we can apply the Midpoint Theorem again to show that PQ is parallel to AC.
Therefore, PQ || AC.
2. Proving DE || PQ:
To show that DE is parallel to PQ, we need to prove that BC is parallel to AC.
Consider triangle AOB. Since D is the midpoint of AB, we can use the Midpoint Theorem to show that DE is parallel to OA (extended). Therefore, DE is parallel to BC.
Now, let's consider triangle AOC. Since Q is the midpoint of OC, we can apply the Midpoint Theorem to show that PQ is parallel to OA (extended). Therefore, PQ is parallel to AC.
Since both pairs of opposite sides are parallel (DE || BC and PQ || AC), we can conclude that DEQP is a parallelogram.
Therefore, the correct answer is option D, a parallelogram.
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D and E are the mid-points of the sides AB and AC ofABC and O is any p...
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D and E are the mid-points of the sides AB and AC ofABC and O is any point on the side BC, O is joined to A. If P and Q are the mid-points of OB and OC res, Then DEQP isa)A Rectangleb)A Trianglec)A Rhombusd)A ParallelogramCorrect answer is option 'D'. Can you explain this answer?
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D and E are the mid-points of the sides AB and AC ofABC and O is any point on the side BC, O is joined to A. If P and Q are the mid-points of OB and OC res, Then DEQP isa)A Rectangleb)A Trianglec)A Rhombusd)A ParallelogramCorrect answer is option 'D'. Can you explain this answer? for Class 9 2025 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about D and E are the mid-points of the sides AB and AC ofABC and O is any point on the side BC, O is joined to A. If P and Q are the mid-points of OB and OC res, Then DEQP isa)A Rectangleb)A Trianglec)A Rhombusd)A ParallelogramCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Class 9 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for D and E are the mid-points of the sides AB and AC ofABC and O is any point on the side BC, O is joined to A. If P and Q are the mid-points of OB and OC res, Then DEQP isa)A Rectangleb)A Trianglec)A Rhombusd)A ParallelogramCorrect answer is option 'D'. Can you explain this answer?.
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