In a cyclic quadrilateral ABCD;a,b,c,ddenote the length of the sides AB,BC,CD,andDA respectively.
Then cosA is equal to.
a2+d2-b2-c22bc+ad
a2+d2-b2-c22bc-ad
a2-d2+b2-2c2bc+ad
none of these
Explanation for the correct answer:
To find the value of cosA:
From the above figure,
Consider triangle ABD,
BD2=a2+d2-2adcosA...1Bycosinerule
Similarly from triangle BDC,
BD2=b2+c2-2bccosC⇒BD2=b2+c2-2bccos180-A[∵ABCDiscyclic,A+C=1800]⇒a2+d2-2adcosA=b2+c2+2bccosA[∵from1]⇒cosA=a2+d2-b2-c22bc+ad
Hence, the correct option is A.
Question 4 P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD such that AC⊥BD. Prove that PQRS is a rectangle.