Advertisements
Advertisements
प्रश्न
Show that the bisectors of angles of a parallelogram form a rectangle
Advertisements
उत्तर
Given: A parallelogram in which bisector of angle A, B, C, D intersect at P, Q, R, S to form a quadrilateral PQRS.
To prove: Quadrilateral PQRS is a rectangle.
Proof: Since ABCD is a parallelogram.
Therefore, AB || DC.
Now, AB || DC, and transversal AD cuts them, so we have
∠A + ∠D = 180°
`1/2 ∠"A" + 1/2 ∠ "D" = (180^circ)/2`
∠DAS + ∠ADS = 90°
But in ΔASD, we have
∠ADS + ∠DAS + ∠ASD = 180°
90° + ∠ASD = 180°
∠ASD = 90°
∠RSP = ∠ASD ...(vertically opposite angle)
∠RSP = 90°
Similarly, we can prove that
∠SRQ = 90°, ∠RQP = 90° and ∠QPS = 90°
Thus, PQRS is a quadrilateral each of whose angle is 90°.
Hence, PQRS is a rectangle.
APPEARS IN
संबंधित प्रश्न
Name the quadrilaterals whose diagonals are equal
Which of the following statement is true for a rectangle?
It has all its sides of equal length.
Which of the following statement true for a square?
Its diagonals bisect each other at right angle.
A mason has made a concrete slab. He needs it to be rectangular. In what different ways can he make sure that it is rectangular?
If the diagonals of a parallelogram are of equal lengths, the parallelogram is a rectangle. Prove it.
ABCD is a rectangle whose diagonals AC and BD intersect at O. If ∠OAB = 46°, find ∠OBC
Rectangle is a regular quadrilateral.
Every parallelogram is a rectangle.
In a rectangle ABCD, AB = 25 cm and BC = 15. In what ratio does the bisector of ∠C divide AB?
A line l is parallel to line m and a transversal p intersects them at X, Y respectively. Bisectors of interior angles at X and Y interesct at P and Q. Is PXQY a rectangle? Given reason.
