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, 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.

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

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

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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, 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.

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

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, 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.

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

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.