Advertisements
Advertisements
प्रश्न
Diagonals of a parallelogram intersect each other at point O. If AO = 5, BO = 12 and AB = 13 then show that `square`ABCD is a rhombus.
उत्तर
Given: AO = 5, BO = 12 and AB = 13
To Prove: `square`ABCD is a rhombus.
Proof:
AO2 + BO2
= 52 + 122
= 25 + 144
= 169 ...(i)
∴ AB2 = 132 = 169 ...(ii)
∴ AB2 = AO2 + BO2 ...[From (i) and (ii)]
∴ ∆AOB is a right-angled triangle. ...[Converse of Pythagoras theorem]
∴ ∠AOB = 90°
∴ seg AC ⊥ seg BD ...(iii) [A-O-C]
∴ In parallelogram ABCD,
∴ seg AC ⊥ seg BD ...[From (iii)]
∴ `square`ABCD is a rhombus. ...[A parallelogram is a rhombus perpendicular to each other]
APPEARS IN
संबंधित प्रश्न
Name the quadrilaterals whose diagonals are perpendicular bisectors of each other
Given below is a parallelogram ABCD. Complete each statement along with the definition or property used.
(i) AD =
(ii) ∠DCB =
(iii) OC =
(iv) ∠DAB + ∠CDA =
The following figure is parallelogram. Find the degree values of the unknowns x, y, z.
The following figure is parallelogram. Find the degree values of the unknown x, y, z.
In a parallelogram ABCD, ∠D = 135°, determine the measure of ∠A and ∠B.
Fill in the blank, in each of the following, so as to make the statement true:
If the diagonals of a parallelogram bisect each other at right angles, then it is a ......
The diagonals of a parallelogram are not perpendicular. Is it a rhombus? Why or why not?
The diagonals of a quadrilateral are of lengths 6 cm and 8 cm. If the diagonals bisect each other at right angles, what is the length of each side of the quadrilateral?
State with reason whether the following statement is ‘true’ or ‘false’.
Every rhombus is a rectangle.
Construct a rhombus whose side is 5 cm and one angle is of 60°.
