A translation maps a point (1,2) onto (-1,2). What would be the coordinates of the object whose image is (-3,-3) under the same translation?

Haya hapa masuluhisho ya maswali uliyotoa:

A translation maps a point $(1,2)$ onto $(-1,2)$. What would be the coordinates of the object whose image is $(-3,-3)$ under the same translation?

• Step 1: Determine the translation vector. Let the translation vector be $T = \begin{pmatrix} a \\ b \end{pmatrix}$. A point $(x,y)$ is translated to its image $(x',y')$ by the rule:

  $$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} a \\ b \end{pmatrix}$$

  Given that the point $(1,2)$ is mapped onto $(-1,2)$:

  $$\begin{pmatrix} -1 \\ 2 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} + \begin{pmatrix} a \\ b \end{pmatrix}$$

  From the x-coordinates: $-1 = 1 + a \implies a = -1 - 1 = -2$. From the y-coordinates: $2 = 2 + b \implies b = 2 - 2 = 0$. So, the translation vector is $T = \begin{pmatrix} -2 \\ 0 \end{pmatrix}$.

• Step 2: Find the coordinates of the object. Let the object's coordinates be $(x_o, y_o)$. Its image is given as $(-3,-3)$. Using the same translation vector:

  $$\begin{pmatrix} -3 \\ -3 \end{pmatrix} = \begin{pmatrix} x_o \\ y_o \end{pmatrix} + \begin{pmatrix} -2 \\ 0 \end{pmatrix}$$

  From the x-coordinates: $-3 = x_o - 2 \implies x_o = -3 + 2 = -1$. From the y-coordinates: $-3 = y_o + 0 \implies y_o = -3$. The coordinates of the object are $(-1,-3)$.

The coordinates of the object are $\boxed{(-1,-3)}$.

17. A bag contains 8 red balls and 6 blue balls all identical in size and shape. Jane selected 3 balls at random without replacement.

(a) Draw a tree diagram to represent this information.

• Step 1: Identify initial probabilities. Total number of balls = 8 Red + 6 Blue = 14 balls. Probability of drawing a Red ball first (R1) = $\frac{8}{14}$ Probability of drawing a Blue ball first (B1) = $\frac{6}{14}$

• Step 2: Draw the tree diagram for three selections without replacement.

                                      Start
                                        |
             ___________________________|___________________________
            /                                                       \
           /                                                         \
      (8/14) R1                                                 (6/14) B1
         /                                                           \
        /                                                             \
       /                                                               \
      /                                                                 \
     /                                                                   \
    /___________________________                                  ___________________________\
   /                             \                               /                             \
  /                               \                             /                               \
(7/13) R2                       (6/13) B2                   (8/13) R2                       (5/13) B2
 /       \                       /       \                   /       \                       /       \
/         \                     /         \                 /         \                     /         \
R3 (6/12) B3 (6/12)           R3 (7/12) B3 (5/12)         R3 (7/12) B3 (5/12)           R3 (8/12) B3 (4/12)