4 A point 'O' is taken inside a rhombus PQRS such that its distances form the angular points P - Maths -

Question

4 A point 'O' is taken inside a rhombus PQRS such that its
distances form the angular points P and R are equal Prove that OQ
and OS are in one and the same straight line

Lovina Kansal · Accepted Answer

Dear student
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Given : A  point O is taken inside a rhombus PQRS OP=ORTo prove: OQ and OS are in one and the same straight line
In △QOP and △QORQP=QR    givenOQ=OQ     common  sideOR=OP  given△QOP ≅ △QOR  SSS∠QOR=∠QOP    CPCT   
1Similarly ∠SOR=∠SOP 
2Now, ∠QOP+∠POS+∠SOR+∠ROQ=360°                                                                 (angular measure around a point)2∠QOP+∠SOP=360°      from 1 and 2    ∠QOP+∠SOP=180°   Here ∠QOP and ∠SOP     forms a linear pair  of anglesand QO and OS are their outer arms which are in the  same straight line/

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4. A point 'O' is taken inside a rhombus PQRS such that its distances form the angular points P and R are equal. Prove that OQ and OS are in one and the same straight line.