Advertisements
Advertisements
प्रश्न
Prove that the figure obtained by joining the mid-points of the adjacent sides of a rectangle is a rhombus.
Advertisements
उत्तर
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
संबंधित प्रश्न
In the given figure, points X, Y, Z are the midpoints of side AB, side BC and side AC of ΔABC respectively. AB = 5 cm, AC = 9 cm and BC = 11 cm. Find the length of XY, YZ, XZ.
In the given figure, ΔABC is an equilateral traingle. Points F, D and E are midpoints of side AB, side BC, side AC respectively. Show that ΔFED is an equilateral traingle.
In triangle ABC, D and E are points on side AB such that AD = DE = EB. Through D and E, lines are drawn parallel to BC which meet side AC at points F and G respectively. Through F and G, lines are drawn parallel to AB which meets side BC at points M and N respectively. Prove that: BM = MN = NC.
If the quadrilateral formed by joining the mid-points of the adjacent sides of quadrilateral ABCD is a rectangle,
show that the diagonals AC and BD intersect at the right angle.
In ΔABC, D, E, F are the midpoints of BC, CA and AB respectively. Find DE, if AB = 8 cm
In ΔABC, D, E and F are the midpoints of AB, BC and AC.
Show that AE and DF bisect each other.
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"`.
The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is ______.
Show that the quadrilateral formed by joining the mid-points of the consecutive sides of a square is also a square.
D, E and F are respectively the mid-points of the sides AB, BC and CA of a triangle ABC. Prove that by joining these mid-points D, E and F, the triangles ABC is divided into four congruent triangles.
