Advertisements
Advertisements
प्रश्न
In the given figure, if points P, Q, R, S are on the sides of parallelogram such that AP = BQ = CR = DS then prove that `square`PQRS is a parallelogram.
Advertisements
उत्तर
Given: `square` ABCD is a parallelogram.
AP = BQ = CR = DS
To prove: `square`PQRS is a parallelogram.
Proof:
AP = CR ...(Given) ...(i)
`square`ABCD is a parallelogram.
AB = CD ...(opposite sides of parallelogram)
∴ AP + PB = CR + RD ...[A-P-B, D-R-C] ...(ii)
∴ PB = RD ...[From (i) and (ii)] ...(iii)
∠ABC ≅ ∠ADC ...(Opposite angles of parallelogram)
That is, ∠PBQ ≅ ∠RDS ...(A-P-B, B-Q-C, C-R-D and A-S-D) ...(iv)
In ΔPBQ and ΔRDS,
Seg PB ≅ Seg RD ...[From (iii)]
∠PBQ ≅ ∠RDS ...[From (iv)]
Seg BQ ≅ Seg SD ...(Given)
∴ ΔPBQ ≅ ΔRDS ...[SAS test]
Seg PQ ≅ Seg RS ...(c.s.c.t) ...(v)
Thus, we can prove that, ΔPAS ≅ ΔRCQ,
∴ Seg PS ≅ Seg RQ ...(vi)
In `square`PQRS,
Seg PQ ≅ Seg RS ...[From (v)]
Seg PS ≅ Seg RQ ...[from (vi)]
If the opposite sides of a quadrilateral are congruent, then it is a parallelogram.
∴ `square`PQRS is a parallelogram.
APPEARS IN
संबंधित प्रश्न
Consider the given parallelogram. Find the values of the unknowns x, y, z.
Consider the given parallelogram. Find the values of the unknowns x, y, z.
Consider the given parallelograms. Find the values of the unknowns x, y, z.
The following figure GUNS is a parallelogram. Find x and y. (Lengths are in cm)
Name the quadrilaterals whose diagonals bisect each other
In the given figure, `square`PQRS and `square`ABCR are two parallelograms. If ∠P = 110° then find the measures of all angles of `square`ABCR.
In the given figure, G is the point of concurrence of medians of ΔDEF. Take point H on ray DG such that D-G-H and DG = GH, then prove that `square`GEHF is a parallelogram.
Ratio of two adjacent sides of a parallelogram is 3 : 4, and its perimeter is 112 cm. Find the length of its each side.
Construct ☐ BARC such that l(BA) = l(BC) = 4.2 cm, l(AC) = 6.0 cm, l(AR) = l(CR) = 5.6 cm
ABCD is a parallelogram. What kind of quadrilateral is it if: AC = BD but AC is not perpendicular to BD?
In parallelogram ABCD, X and Y are midpoints of opposite sides AB and DC respectively. Prove that:
(i) AX = YC
(ii) AX is parallel to YC
(iii) AXCY is a parallelogram.
The angle between the two altitudes of a parallelogram through the same vertex of an obtuse angle of the parallelogram is 30°. The measure of the obtuse angle is ______.
The adjacent sides of a parallelogram are 5 cm and 9 cm. Its perimeter is ______.
In the following figure of a ship, ABDH and CEFG are two parallelograms. Find the value of x.
Two angles of a quadrilateral are each of measure 75° and the other two angles are equal. What is the measure of these two angles? Name the possible figures so formed.
A diagonal of a parallelogram bisects an angle. Will it also bisect the other angle? Give reason.
The angle between the two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 45°. Find the angles of the parallelogram.
In parallelogram ABCD, the angle bisector of ∠A bisects BC. Will angle bisector of B also bisect AD? Give reason.
Construct a parallelogram HOME with HO = 6 cm, HE = 4 cm and OE = 3 cm.
Construct a parallelogram when one of its side is 4 cm and its two diagonals are 5.6 cm and 7 cm. Measure the other side.
