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State the coordinates of the mid-points of the line joining (a) (2,3) and (4,7), (b) (-1,3) and (5,4), (c) (29,b) and (9,2b). Given that the point (9,2) divides the segment of the line from A(6,8) to B(x2,y2) in the ratio AP:PB = 3:7. Find the coordinates of B

Mathematics

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Answers

2 parts
1
(3,5)

(2,72)\left(2, \frac{7}{2}\right)

(19,15)
2
(16, -12)

Step-by-step solution

6 steps

Here are the solutions for the given problems.

Part 1· 3 steps

Mid-points of line segments

The formula for the midpoint M(xm,ym)M(x_m, y_m) of a line segment joining (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:

xm=x1+x22,ym=y1+y22x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2}
  • a) For points (2,3)(2,3) and (4,7)(4,7)

  1. Substitute the coordinates into the midpoint formula.

    $$
x_m = \frac{2 + 4}{2} = \frac{6}{2} = 3
$$
$$
y_m = \frac{3 + 7}{2} = \frac{10}{2} = 5
$$
The midpoint is $\boxed{(3,5)}$.
    • b) For points (−1,3)(-1,3) and (5,4)(5,4)

  2. Substitute the coordinates into the midpoint formula.

    $$
x_m = \frac{-1 + 5}{2} = \frac{4}{2} = 2
$$
$$
y_m = \frac{3 + 4}{2} = \frac{7}{2}
$$
The midpoint is $\boxed{\left(2, \frac{7}{2}\right)}$.
    • c) For points (29,5)(29,5) and (9,25)(9,25) (Assuming '2b' is a typo for '25')

  3. Substitute the coordinates into the midpoint formula.

    $$
x_m = \frac{29 + 9}{2} = \frac{38}{2} = 19
$$
$$
y_m = \frac{5 + 25}{2} = \frac{30}{2} = 15
$$
The midpoint is $\boxed{(19,15)}$.

Part 2(16, -12)

Finding coordinates of B using the section formula

Given point A (6,8)(6,8), point P (9,2)(9,2), and the ratio AP:PB = 3:73:7. Let B be (x2,y2)(x_2, y_2).
The section formula for a point P dividing a segment AB in the ratio m:nm:n is:

x=nx1+mx2m+n,y=ny1+my2m+nx = \frac{nx_1 + mx_2}{m+n}, \quad y = \frac{ny_1 + my_2}{m+n}

Here, (x1,y1)=(6,8)(x_1, y_1) = (6,8), (x,y)=(9,2)(x,y) = (9,2), m=3m=3, n=7n=7.

  1. Set up the equation for the x-coordinate.

    9=7(6)+3(x2)3+79 = \frac{7(6) + 3(x_2)}{3+7} 9=42+3x2109 = \frac{42 + 3x_2}{10} 90=42+3x290 = 42 + 3x_2 3x2=90−423x_2 = 90 - 42 3x2=483x_2 = 48 x2=483=16x_2 = \frac{48}{3} = 16

  2. Set up the equation for the y-coordinate.

    2=7(8)+3(y2)3+72 = \frac{7(8) + 3(y_2)}{3+7} 2=56+3y2102 = \frac{56 + 3y_2}{10} 20=56+3y220 = 56 + 3y_2 3y2=20−563y_2 = 20 - 56 3y2=−363y_2 = -36 y2=−363=−12y_2 = \frac{-36}{3} = -12

  3. State the coordinates of B.

    The coordinates of B are (16,−12)\boxed{(16, -12)}.

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