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| 1. |
In the figure , ABCD is a parallelogram. P, Q , R , S are the midpoint of the sides of the parallelogram . Then prove that PQ = RS and QR = PS |
| Answer» <html><body><p><strong>Answer:</strong></p><p></p><p><strong>Step-by-step explanation:</strong></p><p>Given : -</p><p><a href="https://interviewquestions.tuteehub.com/tag/abcd-360998" style="font-weight:bold;" target="_blank" title="Click to know more about ABCD">ABCD</a> is a parallelogram</p><p>P , Q , R & S are the mid points of the sides AB , BC , CD , DA</p><p>Required to <a href="https://interviewquestions.tuteehub.com/tag/prove-593004" style="font-weight:bold;" target="_blank" title="Click to know more about PROVE">PROVE</a> : -</p><p>PQ = RS</p><p>QR = PS</p><p>Theorem used : -</p><p>Mid point theorem</p><p>This states that ;</p><p>The line segments joining the midpoints of the two sides of the triangle is parallel to the third side and also half of the third side .</p><p></p><h2><em><strong>Solution : -</strong></em></h2><p>ABCD is a parallelogram</p><p></p><p>P , Q , R & S are the mid points of the sides AB , BC , CD , DA</p><p>Here,</p><p>AC is a <a href="https://interviewquestions.tuteehub.com/tag/diagonal-950665" style="font-weight:bold;" target="_blank" title="Click to know more about DIAGONAL">DIAGONAL</a> .</p><p>Consider ∆ ABC & ∆ <a href="https://interviewquestions.tuteehub.com/tag/adc-361797" style="font-weight:bold;" target="_blank" title="Click to know more about ADC">ADC</a></p><p>In ∆ ABC & ∆ ADC .</p><p>=> AC = AC ( side )</p><p>[ Reason : Common side ]</p><p>=> AB = CD ( side )</p><p>[ Reason : In a parallelogram, opposite sides are equal ]</p><p>=> AD = BC ( side )</p><p>[ Reason : In a parallelogram , opposite angles are equal ]</p><p>By using S.S.S. Congruency rule</p><p>∆ ABC ∆ADC</p><p>However,</p><p>Similarly, if you want to write easily you can write that in a parallelogram a diagonal divides it into 2 Congruency triangles . so ∆ ABC ∆ADC</p><p>This implies ;</p><p>PQ = ½ AC</p><p>RS = ½ AC</p><p>[ Reason : Mid point theorem ]</p><p><a href="https://interviewquestions.tuteehub.com/tag/since-644476" style="font-weight:bold;" target="_blank" title="Click to know more about SINCE">SINCE</a>, RHS part is equal . Let's equate the LHS part .</p><p>Hence,</p><p>PQ = RS</p><p>Similarly,</p><h2><em><strong>Construction : -</strong></em></h2><p>Draw the diagonal BD in the ||gm PQRS .</p><p>Consider ∆ BCD & ∆ BAD</p><p>In ∆ BCD & ∆ BAD</p><p>=> BD = BD ( side )</p><p>[ Reason : Common side ]</p><p>=> BC = AD ( side )</p><p>[ Reason : Opposite sides are equal ]</p><p>=> CD = AB ( side )</p><p>[ Reason : Opposite sides are equal ]</p><p>By using S.S.S. Congruency rule ;</p><p>∆ BCD ∆BAD</p><p>This implies ;</p><p>QR = ½ BD</p><p>QR = ½ BD PS = ½ BD</p><p>[ Reason : Mid point theorem ]</p><p>Since,</p><p>RHS part is equal . Let's equate the LHS part</p><p>Hence,</p><p>QR = PS</p><p>Hence Proved </p><h2><em><strong>Hope my answer helps you</strong></em></h2><h2><em><strong>Mark me as brainliest</strong></em></h2><h2></h2><h2></h2><p></p></body></html> | |