Class XI Chapter 12 – Introduction to Three Dimensional Geometry Maths ______________________________________________________________________________ Point A divides PQ in the ratio 1:2. Therefore, by section formula, the coordinates of point A are given by 1(10) 2 4 1( 16) 2 2 1(6) 2 4 , , 6, 4, 2 1 2 1 2 1 2 Point B divides PQ in the ratio 2:1. Therefore, by section formula, the coordinates of point B are given by 2(10) 14 2( 16) 12 2(6) 14 , , 8, 10,2 21 21 21 Thus, (6, â€“4, â€“2) and (8, â€“10, 2) are the points that trisect the line segment joining points P (4, 2, â€“6) and Q (10, â€“16, 6). Exercise Miscellaneous Question 1: Three vertices of a parallelogram ABCD are A (3, -1, 2), B (1, 2, -4) and C (-1, 1, 2). Find the coordinates of the fourth vertex. Solution 1: The three vertices of a parallelogram ABCD are given as A (3, â€“1, 2), B (1, 2, â€“4), and C (â€“1, 1, 2). Let the coordinates of the fourth vertex be D (x, y, z). We know that the diagonals of a parallelogram bisect each other. Therefore, in parallelogram ABCD, AC and BD bisect each other. ∴Mid-point of AC = Mid-point of BD 31 **11** 22 x1 y 2 z 4 , , , , 2 2 2 2 2 2 x 1 y 2 z 4 10,2 , , 2 2 2 Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

Class XI Chapter 12 – Introduction to Three Dimensional Geometry Maths ______________________________________________________________________________ x1 y 2 z 4 1, 0, and 2 2 2 2 ⇒ x = 1, y = â€“2, and z = 8 Thus, the coordinates of the fourth vertex are (1, â€“2, 8). Question 2: Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and (6, 0, 0). Solution 2: Let AD, BE, and CF be the medians of the given triangle ABC. Since AD is the median, D is the mid-point of BC. 06 40 00 ∴Coordinates of point D = , , 2 2 2 = (3, 2, 0) 2 2 2 AD= 03 02 60 9 436 49 7 Since BE is the median, E is the mid-point of AC. 06 00 00 Coordinates of point E = , , 3,0,3 2 2 2 2 2 2 BE= 30 04 30 916 9 34 Since CF is the median, F is the mid-point of AB. 00 0 4 60 Coordinates of point F = , , 0,2,3 2 2 2 2 2 2 Length of CF = 60 02 03 36 49 49 7 Thus, the lengths of the medians of ΔABC are 7, 34 , and 7. Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

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