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Find the coordinate of the point which decide the line segment

Mathematics

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2 parts
c
(10,8)
d
(22,29)

Step-by-step solution

6 steps

Here are the solutions for the remaining parts of question 8:

The general formula for the point P(x,y)P(x,y) that divides a line segment joining (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) in the ratio m:nm:n is:
For internal division: P(x,y)=(mx2+nx1m+n,my2+ny1m+n)P(x,y) = \left(\frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n}\right)
For external division: P(x,y)=(mx2−nx1m−n,my2−ny1m−n)P(x,y) = \left(\frac{m x_2 - n x_1}{m - n}, \frac{m y_2 - n y_1}{m - n}\right)

Part c· 3 steps

$(3,4)$ and $(24,16)$ in the ratio $1:2$ internally.

  1. Identify the given values.

    (x1,y1)=(3,4)(x_1, y_1) = (3,4)
    (x2,y2)=(24,16)(x_2, y_2) = (24,16)
    m=1m = 1, n=2n = 2

  2. Calculate the x-coordinate using the internal division formula.

    x=1×24+2×31+2x = \frac{1 \times 24 + 2 \times 3}{1 + 2} x=24+63x = \frac{24 + 6}{3} x=303x = \frac{30}{3} x=10x = 10

  3. Calculate the y-coordinate using the internal division formula.

    y=1×16+2×41+2y = \frac{1 \times 16 + 2 \times 4}{1 + 2} y=16+83y = \frac{16 + 8}{3} y=243y = \frac{24}{3} y=8y = 8

    The coordinate of the point is (10,8)\boxed{(10,8)}.

Part d(22,29)

$(7,8)$ and $(12,15)$ in the ratio $3:2$ alternatively.

Assuming "alternatively" means externally, as it's a common type of division in coordinate geometry problems.

  1. Identify the given values.

    (x1,y1)=(7,8)(x_1, y_1) = (7,8)
    (x2,y2)=(12,15)(x_2, y_2) = (12,15)
    m=3m = 3, n=2n = 2

  2. Calculate the x-coordinate using the external division formula.

    x=3×12−2×73−2x = \frac{3 \times 12 - 2 \times 7}{3 - 2} x=36−141x = \frac{36 - 14}{1} x=22x = 22

  3. Calculate the y-coordinate using the external division formula.

    y=3×15−2×83−2y = \frac{3 \times 15 - 2 \times 8}{3 - 2} y=45−161y = \frac{45 - 16}{1} y=29y = 29

    The coordinate of the point is (22,29)\boxed{(22,29)}.

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