Plzz fast i have exams 2moro. Dont provide any link Q. Prove that 2 r = (n + l - r) Share with your friends Share 0 Neha Sethi answered this Dear student From the property of tangents we know that the length of two tangents drawn to a circle fromthe same external point will be equal.Therefore, we haveMQ=MPLet us denote MP and MQ as xLP=LRLet us denote LP and LR as yNQ=NRLet us denote NQ and NR as zWe have been given that △LMN is a right triangle and BC=l and AB=n.Let us find out LN using pythagoras theorem,we haveLN2=LM2+MN2⇒LN2=l2+n2⇒LN=l2+n2Consider the perimeter of the given triangle.We have,LM+MN+NL=l+m+n Looking at the figure, we can rewrite it as,LP+PM+MQ+QN+NR+RL=l+m+n⇒2(x+y+z)=l+m+n ...(1)Now, consider the side LN of the triangle.LN=m⇒LR+RN=m ⇒y+z=m ...(2)Subtract (2) from (1).We have2(x+y+z)-y-z=l+m+n-m⇒2x+y+z=l+n⇒x=l+n-y-z2⇒z=l+nNow, consider the quadrilateral MPOQ.We haveMP=MQAlso,OP=OQ (radius)It is given that ∠PMQ=90°From the property of tangenst,we know that the tangent will be at right angle to the radius of the circle at the point of contact.Therefore,∠OPM=90° and ∠OQM=90°We know that sum of all angles of a quadrilateral will be equal to 360.Therefore,∠PMQ+∠OPM+∠OQM+∠POQ=360°⇒90°+90°+90°+∠POQ=360°⇒∠POQ=90°Since all the angles of the quadrilateral are equal to 90° and the adjacent sides are also equal.So, this a square.So,⇒r=MP=l+n-y-z2⇒2r=l+n-m ∵y+z=mPlease recheck your proving part Regards 0 View Full Answer
