Advertisements
Advertisements
Question
ABCD is a rectangle in which diagonal BD bisects ∠B. Show that ABCD is a square.
Advertisements
Solution
Given: In a rectangle ABCD, diagonal BD bisects ∠B.
Construct: Join AC.
To show: ABCD is a square.
Proof: In ΔBAD and ΔBCD,
∠ABD = ∠CBD ...[Given]
∠A = ∠C ...[Each 90°]
And BD = BD ...[Common side]
∴ ΔBAD ≅ ΔBCD ...[By AAS congruence rule]
∴ AB = BC
And AD = CD [By CPCT rule] ...(i)
But in rectangle ABCD, opposite sides are equal.
∴ AB = CD
And BC = AD ...(ii)
From equations (i) and (ii),
AB = BC = CD = DA.
So, ABCD is a square.
Hence proved.
APPEARS IN
RELATED QUESTIONS
Diagonals of a parallelogram `square`WXYZ intersect each other at point O. If ∠XYZ = 135° then what is the measure of ∠XWZ and ∠YZW?
If l(OY)= 5 cm then l(WY)= ?
The diagonals AC and BD of a parallelogram ABCD intersect each other at the point O. If ∠DAC = 32º and ∠AOB = 70º, then ∠DBC is equal to ______.
Diagonals AC and BD of a parallelogram ABCD intersect each other at O. If OA = 3 cm and OD = 2 cm, determine the lengths of AC and BD.
Diagonals of a parallelogram are perpendicular to each other. Is this statement true? Give reason for your answer.
E and F are points on diagonal AC of a parallelogram ABCD such that AE = CF. Show that BFDE is a parallelogram.
A diagonal of a parallelogram bisects one of its angles. Show that it is a rhombus.
P is the mid-point of the side CD of a parallelogram ABCD. A line through C parallel to PA intersects AB at Q and DA produced at R. Prove that DA = AR and CQ = QR.
If the diagonals of a quadrilateral bisect each other, it is a ______.
The point of intersection of diagonals of a quadrilateral divides one diagonal in the ratio 1:2. Can it be a parallelogram? Why or why not?
