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Answers to NCERT/CBSE MATHEMATICS Class 10(Class XI)textbook OPTIONAL. EXERCISE. CHAPTER SEVEN. COORDINATE GEOMETRY. EXERCISES. 7.
NCERT/CBSE MATHEMATICS CLASS 10 textbook(OPTIONAL EXERCISE) http://www.TutorBreeze.com Contact for Online Tutoring in Physics, Math, Chemistry, English Answers to NCERT/CBSE MATHEMATICS Class 10(Class XI)textbook OPTIONAL EXERCISE CHAPTER SEVEN COORDINATE GEOMETRY EXERCISES

7. Let A (4, 2), B(6, 5) and C(1, 4) be the vertices of ∆ ABC. (i) The median from A meets BC at D. Find the coordinates of the point D. SOLUTION:

We know that, The mid-point of a line segment divides the line segment in the ratio 1 : 1. Therefore, the coordinates of the mid-point P of the join of the points A(x1, y1), B(x2, y2) are

Therefore, the coordinates of the mid-point D of the join of the points A(6,5), B(1,4) are  6 +1 5 + 4   7 9  ,  = ,  2  2 2  2

(ii)Find the coordinates of the point P on AD such that AP : PD = 2 : 1

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We know that, the coordinates of the point P(x, y) which divides the line segment joining the points A(x1, y1), B(x2, y2) internally, in the ratio m1 : m2 are

Here P divides AB in the ratio 2 : 1, So, the coordinates of the point P are 7 9    2 × 2 + 1× 4 2 × 2 + 1× 2   7 + 4 9 + 2   11 11  , ,  = = ,  2 + 1 2 + 1 3 3   3 3    

(i) Find the coordinates of points Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1. Now, E is the mid point of AC so the coordinates of E are

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 4 +1 2 + 4   5  , E  =  , 3 2  2   2 BQ : QE = 2 : 1 5    2 × 2 + 1× 6 2 × 3 + 1× 5   5+6 6+5  11 11  ⇒ Q , ,  = Q  = Q ,  2 +1 2 +1  3   3 3 3   

Now, F is the mid point of AB so the coordinates of F are  4+6 2+5  7 , F  =  5,  2   2  2 CR : RF = 2 : 1 7    2 × 5 + 1× 1 2 × 2 + 1× 4   10 + 1 1 + 4   11 11  ⇒ R , ,  = R  = R ,  2 +1  3   3 3 3  2 +1  

(ii) What do you observe? We observe that P, Q and R are the same point. ⇒The three medians pass through the same point ⇒The three medians are concurrent We know that the point of concurrency of the medians of a triangle is called the centroid. (v) If A(x1, y1), B(x2, y2) and C(x3, y3) are the vertices of ∆ ABC, find the coordinates of the centroid of the triangle.

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Consider a triangle ABC with vertices A(x1, y1), B(x2, y2) and C(x3, y3) . Let AD , BE and CF be the medians to the sides BC , AC and AB respectively. i.e D, E , F are the mid points of BC , AC and AB respectively So the coordinates of the points D , E and F are given by the mid – point formula respectively as :  x + x y + y3   x1 + x3 y1 + y3   x1 + x2 y1 + y2  D 2 3 , 2 , ,  ; E  , F  2   2 2   2 2   2 G, the centroid divides AD in the ratio 2:1 Using the section formula the coordinates of G are x2 + x3 y2 + y3    2 × 2 + 1× x1 2 × 2 + 1× y1   x1 + x2 + x3 y1 + y2 + y3  , ,  =  3 3 3 3      

Thus If A(x1, y1), B(x2, y2) and C(x3, y3) are the vertices of ∆ ABC, then the coordinates of the centroid of the triangle are ©TutorBreeze.com Please do not copy the answer given here Write to us for help in understanding the solution

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 x1 + x2 + x3 y1 + y2 + y3  ,   3 3  

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