1
Question
If P and Q are the midpoints of the diagonals AC and BD respectively of a quadrilateral ABCD, prove that ¯¯¯¯¯¯¯¯AB+¯¯¯¯¯¯¯¯¯AD+¯¯¯¯¯¯¯¯CB+¯¯¯¯¯¯¯¯¯CD=4¯¯¯¯¯¯¯¯PQ.
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Solution
We have,
P and Q are the midpoints of diagonal AC and BD of quadrilateral.
Then,
Prove that,
−−→AB+−−→AD+−−→CB+−−→CD=4−−→PQ
Proof:-
Since Q is the mid point of diagonals AC
Then,
−−→AB+−−→AD=2−−→AQ......(1)
Similarly
−−→CB+−−→CD=2−−→CQ......(2)
On adding (1) and (2) to, we get,
−−→AB+−−→AD+−−→CB+−−→CD=2−−→AQ+2−−→CQ
⇒−−→AB+−−→AD+−−→CB+−−→CD=2(−−→AQ+−−→CQ)
⇒−−→AB+−−→AD+−−→CB+−−→CD=−2(−−→QA+−−→QC)
⇒−−→AB+−−→AD+−−→CB+−−→CD=−2(2−−→QP)
⇒−−→AB+−−→AD+−−→CB+−−→CD=−2(−2−−→PQ)
⇒−−→AB+−−→AD+−−→CB+−−→CD=4−−→PQ
Hence proved.
Hence, this is the answer.
We have,
P and Q are the midpoints of diagonal AC and BD of quadrilateral.
Then,
Prove that,
−−→AB+−−→AD+−−→CB+−−→CD=4−−→PQ
Proof:-
Since Q is the mid point of diagonals AC
Then,
−−→AB+−−→AD=2−−→AQ......(1)
Similarly
−−→CB+−−→CD=2−−→CQ......(2)
On adding (1) and (2) to, we get,
−−→AB+−−→AD+−−→CB+−−→CD=2−−→AQ+2−−→CQ
⇒−−→AB+−−→AD+−−→CB+−−→CD=2(−−→AQ+−−→CQ)
⇒−−→AB+−−→AD+−−→CB+−−→CD=−2(−−→QA+−−→QC)
⇒−−→AB+−−→AD+−−→CB+−−→CD=−2(2−−→QP)
⇒−−→AB+−−→AD+−−→CB+−−→CD=−2(−2−−→PQ)
⇒−−→AB+−−→AD+−−→CB+−−→CD=4−−→PQ
Hence proved.
Hence, this is the answer.
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