Geometry

In the following diagram, draw the image of quadrilateral ABCD when translated by the directed line segment v. Label the image of A as A', the image of B as B', the image of C as C', and the image of D as D'.

Step-by-step solution with explanation

Final Answer

Add the horizontal and vertical components of vector v to each vertex: A'=(Ax+Δx, Ay+Δy), B'=(Bx+Δx, By+Δy), C'=(Cx+Δx, Cy+Δy), D'=(Dx+Δx, Dy+Δy). Plot and connect A'→B'→C'→D'→A' to complete the image quadrilateral. (Exact coordinates depend on the specific values of v and ABCD shown in your diagram.)

Step-by-step solution

1

Understand what a translation does

by moving every point the same distance and direction as
A translation slides every point of a shape by the exact same amount. The directed line segment v tells us how far and in which direction to move each vertex.
2

Read the vector v from the diagram

Look at the directed line segment v in your diagram. Count how many units it moves horizontally (right is positive, left is negative) and vertically (up is positive, down is negative). These are your translation values.
3

Translate vertex A to find A'

Take the coordinates of point A and add the horizontal change and vertical change from vector v. Plot the new point and label it A'.
4

Translate vertex B to find B'

Repeat the same process for point B. Add the same vector components to B's coordinates and plot B'.
5

Translate vertices C and D to find C' and D'

Apply the exact same shift to points C and D. Every vertex moves the same way — that is what makes it a translation and not a rotation or reflection.
6

Connect the image vertices in order

Draw line segments connecting A' to B', B' to C', C' to D', and D' back to A'. The new quadrilateral A'B'C'D' should look exactly like ABCD — same shape, same size, just shifted.

Understanding this problem

Learning Insight

A translation is a rigid motion, meaning it preserves distance and angle. Every point moves the same vector, so the image is congruent to the original — no stretching, flipping, or rotating happens. The image and original are always parallel and the same size.

Quick Tip

A fast trick: physically place your pencil at each vertex, then move it exactly the length and direction of v. Wherever your pencil lands is the image point. You can also copy the vector arrow starting at each vertex.

Common Mistake

Students often apply the vector correctly to A and B but accidentally use a different shift for C or D — maybe reading the vector backward (going tail to head instead of head to tail). Make sure you use the SAME Δx and Δy for every single vertex.