7.4 Translations and Vectors
Advanced Geometry
Translation
A translation is a transformation that maps all
points to new points so that the distance from
every old point to every new corresponding
point is equal.
Translation
Translation
Sketch a triangle with vertices A(-1, -3), B(1, -1),
and C(-1, 0). Then sketch the image of the
triangle after the translation…
(x, y)  (x – 3, y + 4)
The translation above is in coordinate notation
TRANSLATION
Sketch a parallelogram with vertices R(-4,-1),
S(-2,0), T(-1,3), U(-3,2). Then sketch the image
of the parallelogram after translation…
(x,y)  (x + 4, y – 2)
Translation
Triangle ABC  Triangle A’B’C’ by a translation
defined by (x, y)  (x – 5, y). The coordinates of
the vertices of triangle ABC are A(7, 4), B(-1, -1),
and C(3, -5). What are the coordinates of the
vertices of triangle A’B’C’?
Vectors
A vector is a quantity that has both direction
and magnitude (size), and is represented by an
arrow drawn between two points.
Component Form
The component form of a vector combines the
horizontal and vertical components.
Let’s take a look at an example….
Identifying Vector Components
The initial point of a vector is V(-2, 3) and the
terminal point is W(-4, -7). Name the vector and
write its component parts.
Identifying Vector Components
The initial point of a vector is E(2, -6) and the
terminal point is F(2, -9). Name the vector and
write its component parts.
Translation Using Vectors
The component form of vector RS is <2,-3>.
Use vector RS to translate the quadrilateral
whose vertices are G(-3,5), H(0,3), J(1,3), and
K(3,-2).
Translation Using Vectors
The component form of vector MN is <3,1>.
Use vector MN to translate the triangle whose
vertices are R(0,4), S(3,1), and T(4,-2).
Translation Using Vectors
The component form of GH is <4, 2>. Use GH to
translate the triangle whose vertices are (3, -1),
B(1, 1), and C(3, 5).
Finding Vectors
In the diagram, ABC maps onto A’B’C’ by a
translation. Write the component form of the
vector that can be used to describe the
translation.
Finding Vectors
In the diagram, EFGH maps onto E’F’G’H’ by a
translation. Write the component form of the
vector that can be used to describe the
translation.
Finding Vectors
In the diagram, ABC maps onto A’B’C’ by a
translation. Write the component form of the
vector that can be used to describe the
translation.
Using Vectors
The coordinates of a logging site are S(5, 7). A logging
truck traveling a straight line from the site to a mill at M
encounters a road detour at D when it is 4 mi west and 2
mi south of the logging site. The truck must travel an
alternate route to A(4, 3).
a. Write the component forms of the vectors from S to D
and from D to A.
b. The mill is 6 mi west and 3 mi south of the logging site.
Write the component form of the vector that describes
the route the logging truck can follow to arrive at the
mill.