A triangle with vertices A(1, 1), B(2, 3) and C(4, 1) is mapped onto its image by a translation T = [−4, 1]…

Question

A triangle with vertices A(1, 1), B(2, 3) and C(4, 1) is mapped onto its image by a translation T = [−4, 1]. Find the coordinates of the images of triangle ABC.

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for ITEM 3 and ITEM 4.

ITEM 3
Some S.2 students of Taibah international school obtained the following marks in a Math test.

Name	Grade
Daniel	80
Tendo	78
Christopher	84
Hannah	80
Nicole	92
Joshua	84

a) Illustrate the mapping of marks to students.
To illustrate the mapping, we show the relationship from the set of students to the set of grades.

Students $\to$ Grades

Daniel $\to$ 80
Tendo $\to$ 78
Christopher $\to$ 84
Hannah $\to$ 80
Nicole $\to$ 92
Joshua $\to$ 84

b) What type of mapping is this?
This is a many-to-one mapping. This is because multiple students (e.g., Daniel and Hannah) map to the same grade (80), and multiple students (e.g., Christopher and Joshua) map to the same grade (84).

ITEM 4

a) A triangle with vertices $A(1, 1)$ , $B(2, 3)$ and $C(4, 1)$ is mapped onto its image by a translation $T = \begin{pmatrix} -4 \\ 1 \end{pmatrix}$ . Find the coordinates of the images of triangle ABC.

To find the image of a point $(x, y)$ under a translation $T = \begin{pmatrix} h \\ k \end{pmatrix}$ , the new coordinates $(x', y')$ are given by $(x+h, y+k)$ .

Find the image of vertex A.
For $A(1, 1)$ and $T = \begin{pmatrix} -4 \\ 1 \end{pmatrix}$ :
$A' = (1 + (-4), 1 + 1) = (1 - 4, 2) = \boxed{(-3, 2)}$
Find the image of vertex B.
For $B(2, 3)$ and $T = \begin{pmatrix} -4 \\ 1 \end{pmatrix}$ :
$B' = (2 + (-4), 3 + 1) = (2 - 4, 4) = \boxed{(-2, 4)}$
Find the image of vertex C.
For $C(4, 1)$ and $T = \begin{pmatrix} -4 \\ 1 \end{pmatrix}$ :
$C' = (4 + (-4), 1 + 1) = (4 - 4, 2) = \boxed{(0, 2)}$
b) Evaluate the following. If $a = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$ and $b = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$ , find:

(i) $2a$
Multiply the vector $a$ by the scalar 2.
$2a = 2 \begin{pmatrix} 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 \times 4 \\ 2 \times 2 \end{pmatrix} = \boxed{\begin{pmatrix} 8 \\ 4 \end{pmatrix}}$
(ii) $2a + 3b$
Calculate $2a$.
From part (i), we know:
$2a = \begin{pmatrix} 8 \\ 4 \end{pmatrix}$
Calculate $3b$.
$3b = 3 \begin{pmatrix} 2 \\ -3 \end{pmatrix} = \begin{pmatrix} 3 \times 2 \\ 3 \times (-3) \end{pmatrix} = \begin{pmatrix} 6 \\ -9 \end{pmatrix}$
Add the resulting vectors $2a$ and $3b$.
$2a + 3b = \begin{pmatrix} 8 \\ 4 \end{pmatrix} + \begin{pmatrix} 6 \\ -9 \end{pmatrix} = \begin{pmatrix} 8 + 6 \\ 4 + (-9) \end{pmatrix} = \begin{pmatrix} 14 \\ 4 - 9 \end{pmatrix} = \boxed{\begin{pmatrix} 14 \\ -5 \end{pmatrix}}$

A triangle with vertices A(1, 1), B(2, 3) and C(4, 1) is mapped onto its image by a translation T = [-4, 1]. Find the coordinates of the images of triangle ABC.

Step-by-step solution

Find the image of vertex A.

Find the image of vertex B.

Find the image of vertex C.

Multiply the vector $a$ by the scalar 2.

Calculate $2a$.

Calculate $3b$.

Add the resulting vectors $2a$ and $3b$.

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