Advertisements
Advertisements
Question
Prove that the figure obtained by joining the mid-points of the adjacent sides of a rectangle is a rhombus.
Advertisements
Solution
Join AC and BC.
In ΔABC, P and Q are the mid-point of AB and BC respectively.
PQ = `(1)/(2)"AC"`.......(i) and PQ || AC
In ΔBDC, R and Q are the mid-points of CD and BC respectively.
QR = `(1)/(2)"BD"`.......(ii) and QR || BD
But AC = BD ...(diagonals of a rectangle)
From (i) and (ii)
PQ = QR
Similarly, QR = RS, RS = SP and RS || AC, SP || BD
Hence, PQ = QR = PS = SP
Therefore, PQRS is a rhombus.
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 a triangle ABC, AD is a median and E is mid-point of median AD. A line through B and E meets AC at point F.
Prove that: AC = 3AF.
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 parallelogram ABCD, E is the mid-point of AB and AP is parallel to EC which meets DC at point O and BC produced at P.
Prove that:
(i) BP = 2AD
(ii) O is the mid-point of AP.
In ΔABC, BE and CF are medians. P is a point on BE produced such that BE = EP and Q is a point on CF produced such that CF = FQ. Prove that: QAP is a straight line.
In ΔABC, P is the mid-point of BC. A line through P and parallel to CA meets AB at point Q, and a line through Q and parallel to BC meets median AP at point R. Prove that: AP = 2AR
ABCD is a parallelogram.E is the mid-point of CD and P is a point on AC such that PC = `(1)/(4)"AC"`. EP produced meets BC at F. Prove that: F is the mid-point of BC.
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 the given figure, PS = 3RS. M is the midpoint of QR. If TR || MN || QP, then prove that:
ST = `(1)/(3)"LS"`
E is the mid-point of a median AD of ∆ABC and BE is produced to meet AC at F. Show that AF = `1/3` AC.
