Advertisements
Advertisements
Question
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.
Advertisements
Solution
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
RELATED QUESTIONS
The following figure is parallelogram. Find the degree values of the unknown x, y, z.
Can the following figure be parallelogram. Justify your answer.
In the following figure RISK and CLUE are parallelograms. Find the measure of x.
ABCD is a parallelogram in which ∠A = 70°. Compute ∠B, ∠C and ∠D.
The sum of two opposite angles of a parallelogram is 130°. Find all the angles of the parallelogram.
All the angles of a quadrilateral are equal to each other. Find the measure of each. Is the quadrilateral a parallelogram? What special type of parallelogram is it?
The perimeter of a parallelogram is 150 cm. One of its sides is greater than the other by 25 cm. Find the length of the sides of the parallelogram.
Find the angles marked with a question mark shown in Fig. 17.27
Which of the following statement is true for a rhombus?
It has all its sides of equal lengths.
Fill in the blank, in the following, so as to make the statement true:
The diagonals of a rhombus ...... each other at ...... angles.
Construct a rhombus whose diagonals are of length 10 cm and 6 cm.
Show that each diagonal of a rhombus bisects the angle through which it passes.
Which of the following statement is true for a rectangle?
Its diagonals are equal and bisect each other.
Measure of one angle of a rhombus is 50°, find the measures of remaining three angles.
The given figure shows a rhombus ABCD in which angle BCD = 80°. Find angles x and y.
The lengths of the diagonals of a Rhombus are 12 cm and 16 cm. Find the side of the rhombus
If the adjacent sides of a parallelogram are equal then parallelogram is a ______.
All rhombuses are squares.
ABCD is a rhombus such that the perpendicular bisector of AB passes through D. Find the angles of the rhombus.
Hint: Join BD. Then ∆ABD is equilateral.
