Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
Consider the given parallelograms. Find the values of the unknowns x, y, z.
Can a quadrilateral ABCD be a parallelogram if ∠D + ∠B = 180°?
Can a quadrilateral ABCD be a parallelogram if AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?
The following figure GUNS is a parallelogram. Find x and y. (Lengths are in cm)
In the given figure, `square`ABCD is a parallelogram. Point E is on the ray AB such that BE = AB then prove that line ED bisects seg BC at point F.
Ratio of two adjacent sides of a parallelogram is 3 : 4, and its perimeter is 112 cm. Find the length of its each side.
In the adjacent figure, if seg AB || seg PQ, seg AB ≅ seg PQ, seg AC || seg PR, seg AC ≅ seg PR then prove that, seg BC || seg QR and seg BC ≅ seg QR.
Construct ☐ PQRS, such that l(PQ) = 3.5 cm, l(QR) = 5.6 cm, l(RS) = 3.5 cm, m∠Q = 110°, m∠R = 70°. If it is given that ☐ PQRS is a parallelogram, which of the given information is unnecessary?
Construct ☐ BARC such that l(BA) = l(BC) = 4.2 cm, l(AC) = 6.0 cm, l(AR) = l(CR) = 5.6 cm
Ratio of consecutive angles of a quadrilateral is 1 : 2 : 3 : 4. Find the measure of its each angle. Write, with reason, what type of a quadrilateral it is.
In parallelogram ABCD, ∠A = 3 times ∠B. Find all the angles of the parallelogram. In the same parallelogram, if AB = 5x – 7 and CD = 3x +1 ; find the length of CD.
ABCD is a parallelogram. What kind of quadrilateral is it if: AC = BD but AC is not perpendicular to BD?
In parallelogram ABCD, E is the mid-point of side AB and CE bisects angle BCD. Prove that:
- AE = AD,
- DE bisects and ∠ADC and
- Angle DEC is a right angle.
In the following diagram, the bisectors of interior angles of the parallelogram PQRS enclose a quadrilateral ABCD.
Show that:
(i) ∠PSB + ∠SPB = 90°
(ii) ∠PBS = 90°
(iii) ∠ABC = 90°
(iv) ∠ADC = 90°
(v) ∠A = 90°
(vi) ABCD is a rectangle
Thus, the bisectors of the angles of a parallelogram enclose a rectangle.
The given figure shows parallelogram ABCD. Points M and N lie in diagonal BD such that DM = BN.
Prove that:
(i) ∆DMC = ∆BNA and so CM = AN
(ii) ∆AMD = ∆CNB and so AM CN
(iii) ANCM is a parallelogram.
In the given figure, AB || EC, AB = AC and AE bisects ∠DAC. Prove that:
- ∠EAC = ∠ACB
- ABCE is a parallelogram.
The adjacent sides of a parallelogram are 5 cm and 9 cm. Its perimeter is ______.
A diagonal of a parallelogram bisects an angle. Will it also bisect the other angle? Give reason.
In the following figure, FD || BC || AE and AC || ED. Find the value of x.
The following figure RUNS is parallelogram. Find x and y. (Lengths are in cm)
