Prove that the quadrilateral obtained by joining the mid-points of an isosceles trapezium is a rhombus.

P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Join AC and BD. Since, ABCD is an isosceles trapezium Its diagonals are equal. AC = BD = x (let) Now, in ∆ABC P and Q are the mid-points of AB and BC So, PQ || AC and PQ = AC = x (By midpoint theorem) … (i) Similarly, in ∆ADC S and R mid-point of AD and CD So, SR || AC and SR = AC = x (By midpoint theorem) … (ii) From (i) and (ii), we have PQ || SR and PQ = SR = .......(iii) In ∆CBD, R and Q are the mid-points of CD and BC So, QR || BD and QR = BD = x (By midpoint theorem) … (iv) Similarly, in ∆ABD S and P mid-point of AD and AB So, SP || BD and SP = BD = x (By midpoint theorem) … (v) From (iv) and (v), we have QR || SP and QR = SP = .......(vi) From (iii) and (vi) we get, PQ = QR = SR = SP and QR || SP, PQ || SR. So, sides of PQRS are equal and opposite sides are parallel. Hence, proved that PQRS is a rhombus.

In the adjoining figure, ABCD is a parallelogram. CB is produced to E such that BE = BC. Prove that AEBD is a parallelogram.

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In the adjoining figure, ABC is an isosceles triangle in which AB = AC. AD bisects exterior angle PAC and CD || BA. Show that
(i) ∠DAC = ∠BCA
(ii) ABCD is a parallelogram.

Mathematics

Prove that the quadrilateral obtained by joining the mid-points of an isosceles trapezium is a rhombus.

Rectilinear Figures

Related Questions

In the adjoining figure, ABCD is a parallelogram. CB is produced to E such that BE = BC. Prove that AEBD is a parallelogram.

In the adjoining figure, ABC is an isosceles triangle in which AB = AC. AD bisects exterior angle PAC and CD || BA. Show that
(i) ∠DAC = ∠BCA
(ii) ABCD is a parallelogram.

Find the size of each lettered angle in the following figure.

Find the size of each lettered angle in the following figure.

Mathematics

Prove that the quadrilateral obtained by joining the mid-points of an isosceles trapezium is a rhombus.

Rectilinear Figures

Related Questions

In the adjoining figure, ABCD is a parallelogram. CB is produced to E such that BE = BC. Prove that AEBD is a parallelogram.

In the adjoining figure, ABC is an isosceles triangle in which AB = AC. AD bisects exterior angle PAC and CD || BA. Show that(i) ∠DAC = ∠BCA(ii) ABCD is a parallelogram.

Find the size of each lettered angle in the following figure.

Find the size of each lettered angle in the following figure.

In the adjoining figure, ABC is an isosceles triangle in which AB = AC. AD bisects exterior angle PAC and CD || BA. Show that
(i) ∠DAC = ∠BCA
(ii) ABCD is a parallelogram.