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In the school garden, Arun (A), Babu (B), Chandra (C) and Daya (D) planted flower plants of Sunflower, Rose, Champa and Jasmine respectively at point A$(2, 8)$, B$(7, 8)$, C$(9, 3)$ and D$(2, 3)$ respectively.
Based on the above, answer the following questions :
(i) Find the distances AB and AD.
(ii) Find BC – CD.
(iii) (a) If Varun wants to plant his flower plant at a point M such that DM: MC = $3:2$, find the coordinates of M.
OR
(b) If N divides the line segment AC in the ratio $2: 3$, find the coordinates of N.
Based on the above, answer the following questions :
(i) Find the distances AB and AD.
(ii) Find BC – CD.
(iii) (a) If Varun wants to plant his flower plant at a point M such that DM: MC = $3:2$, find the coordinates of M.
OR
(b) If N divides the line segment AC in the ratio $2: 3$, find the coordinates of N.
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Solution:
(i) $AB = \sqrt{(7-2)^2+(8-8)^2} = 5$ (0.5 Mark)
$AD = \sqrt{(2-2)^2+(3-8)^2} = 5$ (0.5 Mark)
(ii) $BC – CD = \sqrt{(9 – 7)^2 + (3 – 8)^2} - \sqrt{(9 – 2)^2 + (3 – 3)^2} = \sqrt{29} – 7$ (1 Mark)
(iii) (a) Let the coordinates of M be $(x, y)$
$x = \frac{3\times9+2\times2}{3+2} = \frac{31}{5}$ (1 Mark)
$y = \frac{3\times3+2\times3}{3+2} = 3$
$\therefore$ Coordinates of M are $(\frac{31}{5}, 3)$ (1 Mark)
OR
(b) Let the coordinates of N be $(x, y)$
$x = \frac{2\times9+3\times2}{2+3} = \frac{24}{5}$ (1 Mark)
$y = \frac{2\times3+3\times8}{2+3} = 6$
$\therefore$ Coordinates of N are $(\frac{24}{5}, 6)$ (1 Mark)
(i) $AB = \sqrt{(7-2)^2+(8-8)^2} = 5$ (0.5 Mark)
$AD = \sqrt{(2-2)^2+(3-8)^2} = 5$ (0.5 Mark)
(ii) $BC – CD = \sqrt{(9 – 7)^2 + (3 – 8)^2} - \sqrt{(9 – 2)^2 + (3 – 3)^2} = \sqrt{29} – 7$ (1 Mark)
(iii) (a) Let the coordinates of M be $(x, y)$
$x = \frac{3\times9+2\times2}{3+2} = \frac{31}{5}$ (1 Mark)
$y = \frac{3\times3+2\times3}{3+2} = 3$
$\therefore$ Coordinates of M are $(\frac{31}{5}, 3)$ (1 Mark)
OR
(b) Let the coordinates of N be $(x, y)$
$x = \frac{2\times9+3\times2}{2+3} = \frac{24}{5}$ (1 Mark)
$y = \frac{2\times3+3\times8}{2+3} = 6$
$\therefore$ Coordinates of N are $(\frac{24}{5}, 6)$ (1 Mark)