Advertisements
Advertisements
Question
The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rhombus, if ______.
Options
PQRS is a rhombus
PQRS is a parallelogram
diagonals of PQRS are perpendicular
diagonals of PQRS are equal
Advertisements
Solution
The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rhombus, if diagonals of PQRS are equal.
Explanation:
Given, the quadrilateral ABCD is a rhombus.
So, sides AB, BC, CD and AD are equal.
Now, in ΔPQS, we have
D and C are the mid-points of PQ and PS.
So, `DC = 1/2 QS` [By mid-point theorem] ...(i)
Similarly, in ΔPSR, `BC = 1/2 PR` [By mid-point theorem] ...(ii)
As BC = DC ...[Since, ABCD is a rhombus]
∴ `1/2 QS = 1/2 PR` ...[From equations (i) and (ii)]
⇒ QS = PR
Hence, diagonals of PQRS are equal.
APPEARS IN
RELATED QUESTIONS
ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F (see the given figure). Show that F is the mid-point of BC.
ABCD is a parallelogram, E and F are the mid-points of AB and CD respectively. GH is any line intersecting AD, EF and BC at G, P and H respectively. Prove that GP = PH.
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 given figure, AD and CE are medians and DF // CE.
Prove that: FB = `1/4` AB.
ΔABC is an isosceles triangle with AB = AC. D, E and F are the mid-points of BC, AB and AC respectively. Prove that the line segment AD is perpendicular to EF and is bisected by it.
In a parallelogram ABCD, E and F are the midpoints of the sides AB and CD respectively. The line segments AF and BF meet the line segments DE and CE at points G and H respectively Prove that: ΔGEA ≅ ΔGFD
The diagonals AC and BD of a quadrilateral ABCD intersect at right angles. Prove that the quadrilateral formed by joining the midpoints of quadrilateral ABCD is a rectangle.
In the given figure, T is the midpoint of QR. Side PR of ΔPQR is extended to S such that R divides PS in the ratio 2:1. TV and WR are drawn parallel to PQ. Prove that T divides SU in the ratio 2:1 and WR = `(1)/(4)"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.
P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD in which AC = BD. Prove that PQRS is a rhombus.
