Question

When translating a figure on a graph, how do you determine it's slope and length?Specifically, with this question how would I determine the image's length when the length of the pre image is a square root?

Answer 1

From the figure, we have the coordinates of ABC:'

A(-5, 1), B(-3, 4), C(0, 1)

Given the rule of the translations:

(x, y) ==> (x + 3, y + 2)

Translation is a form of rigid transformation which just moves the figure. Translation only changes the position of a figure but it does not change the shape of the size of the figure.

Using the translation rule, we have the coordinates of A'B'C':

A'(-5+3, 1+2) ==> A'(-2, 3)

B'(-3+3, 4+2) ==> B'(0, 6)

C'(0+3, 1+2) ==> C'(3, 3)

The coordinates are:

A'(-2, 3), B'(0, 6), C'(3, 3)

This means the triangle ABC moved 3 units up and 2 units to the right.

Let's answer the following:

• (a). If the slope of BC is -1, let's determine the slope of B'C'.

Since translation does not change the shape nor the size, the slope of B'C' will also be -1.

• (b). ,If the length of BC is 3√2, the length of B'C' will also be 3√2. This is because translation does not change the size of an object.

• (c). ,Determine the length of AC and A'C'

Apply the distance formula:

`d=√((x_2-x_1)^2+(y_2-y_1)^2)`

• Length of AC:

Where:

(x1, y1) ==> A(-5, 1)

(x2, y2) ==> B(0, 1)

Thus, we have:

`\begin{gathered} AC=√((0-5)^2+(1-1)^2) \\ \\ AC=√((-5)^2) \\ \\ AC=5 \end{gathered}`

The length of AC is 5.

• Length of A'C'

Where:

(x1, y1) ==> A'(-2, 3)

(x2, y2) ==> C'(3, 3)

We have:

`\begin{gathered} A^(\prime)C^(\prime)=√((3-(-2))^2+(3-3)^2) \\ \\ A^(\prime)C^(\prime)=√((3+2)^2+(0)^2) \\ \\ A^(\prime)C^(\prime)=√((5)^2) \\ \\ A^(\prime)C^(\prime)=5 \end{gathered}`

The length of A'C' is also 5 units.

• (d). Slope of AC and A'C'

AC and A'C' are horizontal lines.

The slope of any horizontal line is always zero.

Therefore the slope of AC and A'C' are 0.

ANSWER:

• (a) Slope of B'C' = -1

,

• (b). Length of B'C' = 3√2

,

• (c). Length of AC = 5

Length of A'C' = 5

• (d). Slope of AC = 0

Slope of A'C' = 0