Advertisements
Advertisements
Question
The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is ______.
Options
a rhombus
a rectangle
a square
any parallelogram
Advertisements
Solution
The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is a rectangle.
Explanation:
The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.
Join AC, RP and SQ
In ∆ABC,
P is midpoint of AB and Q is midpoint of BC
∴ By midpoint theorem,
PQ || AC and PQ = `1/2` AC ...(1)
Similarly,
In ∆DAC,
S is midpoint of AD and R is midpoint of CD
∴ By midpoint theorem,
SR || AC and SR = `1/2` AC ...(2)
From (1) and (2),
PQ || SR and PQ = SR
⇒ PQRS is a parallelogram
ABQS is a parallelogram
⇒ AB = SQ
PBCR is a parallelogram
⇒ BC = PR
⇒ AB = PR ...[∵ BC = AB, sides of rhombus]
⇒ SQ = PR
∴ Diagonals of the parallelogram are equal.
Hence, it is a rectangle.
APPEARS IN
RELATED QUESTIONS
ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.
In below Fig, ABCD is a parallelogram in which P is the mid-point of DC and Q is a point on AC such that CQ = `1/4` AC. If PQ produced meets BC at R, prove that R is a mid-point of BC.
In the adjacent figure, `square`ABCD is a trapezium AB || DC. Points M and N are midpoints of diagonal AC and DB respectively then prove that MN || AB.
In the given figure, M is mid-point of AB and DE, whereas N is mid-point of BC and DF.
Show that: EF = AC.
In the figure, give below, 2AD = AB, P is mid-point of AB, Q is mid-point of DR and PR // BS. Prove that:
(i) AQ // BS
(ii) DS = 3 Rs.
In the given figure, AD and CE are medians and DF // CE.
Prove that: FB = `1/4` AB.
Prove that the straight lines joining the mid-points of the opposite sides of a quadrilateral bisect each other.
In parallelogram ABCD, P is the mid-point of DC. Q is a point on AC such that CQ = `(1)/(4)"AC"`. PQ produced meets BC at R. Prove that
(i) R is the mid-point of BC, and
(ii) PR = `(1)/(2)"DB"`.
In ΔABC, the medians BE and CD are produced to the points P and Q respectively such that BE = EP and CD = DQ. Prove that: A is the mid-point of PQ.
In ∆ABC, AB = 5 cm, BC = 8 cm and CA = 7 cm. If D and E are respectively the mid-points of AB and BC, determine the length of DE.
