In the given figure, A, B, C and D are mid-points of PQ, QR, RS and PS respectively. E, F, G and H are mid-points of AB, BC, CD and AD respectively. Which type of quadrilaterals are ABCD and EFGH?
Mid-point Theorem
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Question

In the given figure, A, B, C and D are mid-points of PQ, QR, RS and PS respectively. E, F, G and H are mid-points of AB, BC, CD and AD respectively. Which type of quadrilaterals are ABCD and EFGH?

KnowledgeBoat · Accepted Answer

Join QS, PR, AC and BD. By mid-point theorem, The line segment joining the mid points of any two sides of a triangle is parallel to the third side and is equal to half of it. In △QRP, A and B are midpoints of PQ and QR respectively. ∴ AB || PR and AB = PR (By midpoint theorem) .....(1) Similarly in △PRS, D and C are midpoints of PS and RS respectively. ∴ DC || PR and DC = PR (By midpoint theorem) .....(2) In △PQS, D and A are midpoints of PS and PQ respectively. ∴ DA || QS and DA = QS (By midpoint theorem) .....(3) Similarly in △QRS, B and C are midpoints of QR and SR respectively. ∴ BC || QS and BC = QS (By midpoint theorem) .....(4) From (1) and (2) we get, AB = DC and AB || DC From (3) and (4) we get, DA = BC and DA || BC Since, opposite sides are parallel and equal. Thus, ABCD is a parallelogram. In △ABC, E and F are midpoints of AB and BC respectively. ∴ EF || AC and EF = AC (By midpoint theorem) .....(5) Similarly in △ADC, H and G are midpoints of AD and CD respectively. ∴ GH || AC and GH = AC (By midpoint theorem) .....(6) In △ABD, H and E are midpoints of AD and AB respectively. ∴ EH || BD and EH = BD (By midpoint theorem) .....(7) Similarly in △BCD, F and G are midpoints of BC and CD respectively. ∴ FG || BD and FG = BD (By midpoint theorem) .....(8) From (5) and (6) we get, EF = GH AND EF || GH From (7) and (8) we get, EH = FG and EH || FG Since, opposite sides are parallel and equal. Thus, EFGH is a parallelogram. Hence, ABCD and EFGH are parallelograms.

Mathematics

In the given figure, A, B, C and D are mid-points of PQ, QR, RS and PS respectively. E, F, G and H are mid-points of AB, BC, CD and AD respectively. Which type of quadrilaterals are ABCD and EFGH?

Mid-point Theorem

Related Questions

D and E are the mid-points of the sides AB and AC respectively of △ABC. DE is produced to F. To show that CF is equal and parallel to DA, we need an additional information, which is :
DE = EF
AE = EF
∠DAE = ∠EFC
∠ADE = ∠ECF

In which of the following cases you will get 2XY = QR for the given figure?
(i) When PX = QX and PY = RY
(ii) When PX = QX and a + b = 180°
Only in case (i)
Only in case (ii)
In both the cases
None of these

In the figure, R is the mid-point of AB, P is the mid-point of AR and L is the mid-point of AP. If RS, PQ and LM are parallel to each other, then the length of BC is :
3 LM
4 LM
6 LM
8 LM

In the given figure, △ABC is a scalene triangle in which ∠B = 90°. P is the mid-point of AB, PQ || BC and QM ⊥ BC. Which type of quadrilateral is PQMB?

Mathematics

In the given figure, A, B, C and D are mid-points of PQ, QR, RS and PS respectively. E, F, G and H are mid-points of AB, BC, CD and AD respectively. Which type of quadrilaterals are ABCD and EFGH?

Mid-point Theorem

Related Questions

D and E are the mid-points of the sides AB and AC respectively of △ABC. DE is produced to F. To show that CF is equal and parallel to DA, we need an additional information, which is :DE = EFAE = EF∠DAE = ∠EFC∠ADE = ∠ECF

In which of the following cases you will get 2XY = QR for the given figure?(i) When PX = QX and PY = RY(ii) When PX = QX and a + b = 180°Only in case (i)Only in case (ii)In both the casesNone of these

In the figure, R is the mid-point of AB, P is the mid-point of AR and L is the mid-point of AP. If RS, PQ and LM are parallel to each other, then the length of BC is :3 LM4 LM6 LM8 LM

In the given figure, △ABC is a scalene triangle in which ∠B = 90°. P is the mid-point of AB, PQ || BC and QM ⊥ BC. Which type of quadrilateral is PQMB?

D and E are the mid-points of the sides AB and AC respectively of △ABC. DE is produced to F. To show that CF is equal and parallel to DA, we need an additional information, which is :
DE = EF
AE = EF
∠DAE = ∠EFC
∠ADE = ∠ECF

In which of the following cases you will get 2XY = QR for the given figure?
(i) When PX = QX and PY = RY
(ii) When PX = QX and a + b = 180°
Only in case (i)
Only in case (ii)
In both the cases
None of these

In the figure, R is the mid-point of AB, P is the mid-point of AR and L is the mid-point of AP. If RS, PQ and LM are parallel to each other, then the length of BC is :
3 LM
4 LM
6 LM
8 LM