A group of class IX students goes to a picnic during winter holidays. The positions of three friends Nitin, Rajesh and Kareem are shown by the points P, Q and R respectively.
(i) Find the distance between :
(a) Nitin and Rajesh
(b) Rajesh and Kareem
(c) Nitin and Kareem
(ii) Show that P, Q and R are collinear.
(iii) Find the point on x-axis which is equidistant from points Q and R.
(iv) If R is taken as origin; what will be the co-ordinates of P and Q ?

Question

A group of class IX students goes to a picnic during winter holidays. The positions of three friends Nitin, Rajesh and Kareem are shown by the points P, Q and R respectively.
(i) Find the distance between :
(a) Nitin and Rajesh
(b) Rajesh and Kareem
(c) Nitin and Kareem
(ii) Show that P, Q and R are collinear.
(iii) Find the point on x-axis which is equidistant from points Q and R.
(iv) If R is taken as origin; what will be the co-ordinates of P and Q ?

KnowledgeBoat · Accepted Answer

Given, The co-ordinates are : P(Nitin): (6, 4) Q(Rajesh): (11, 9) R(Kareem): (9, 7) We know that Distance between the given points = (i) (a) Distance between Nitin(P) and Rajesh(Q), Hence, distance between Nitin and Rajesh = units. (b) Distance between Rajesh(Q) and Kareem(R), Hence, distance between Rajesh and Kareem = units. (c) Distance between Nitin(P) and Kareem(R), Hence, distance between Nitin and Kareem = units. (ii) Points P, Q and R are collinear only if, PR + RQ = PQ Consider L.H.S, PR + RQ = = Consider R.H.S, PQ = ∴ L.H.S = R.H.S So, they are collinear. Hence, P, Q and R are collinear. (iii) Let S (x, 0) be the point on x-axis equidistant from Q and R. According to question, SQ = SR Squaring on both sides, (x - 11)2 + (0 - 9)2 = (x - 9)2 + (0 - 7)2 x2 - 22x + 121 + 81 = x2 - 18x + 81 + 49 -18x + 22x = 121 + 81 - 81 - 49 4x = 121 - 49 4x = 72 x = = 18. ∴ The point S = (18, 0). Hence, point on x-axis that is equidistant from Q and R = (18, 0). (iv) If R is taken as origin, to find points of P and Q, their coordinates must be subtracted from the original coordinates of R(9, 7). P' = (6 - 9, 4 - 7) = (-3, -3) Q' = (11 - 9, 9 - 7) = (2, 2) Hence, coordinates of P and Q are = (-3, -3) and (2, 2) respectively.

Mathematics

Distance Formula

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