# Geometric Transformations and Properties

**Learning Targets**: I can use matrices to perform geometric transformations. I can use matrices to calculate geometric properties of certain shapes.

## Warm-Up

What transformation maps triangle ABC onto triangle (ABC)'?

**First things first...**

**Reflections**Check out the 3 transformation matrices below. Use the GeoGebra applet below to explore each of these reflections by changing the elements in matrix A.

What is the reflection indicated by the matrix ?

**Deformations**Check out some transformation matrices below, then use the GeoGebra applet to better understand each type of deformation by editing the elements in matrix A.

**Rotations (2-D)**

The matrix above is the 2-dimensional rotation matrix, which rotates CCW about the origin. See a demonstration in the applet below.

Do you think the determinant of the rotation matrix is dependent on the angle of rotation?

**Rotations (3-D)**

We can rotate vectors in 3-dimensions too! Look at the 3 rotation matrices above. Each matrix rotates a vector CCW about a specific axis. Which one do you think rotates around the z-axis?

Check out the applet below.

**Triangle Area**We can use matrices to calculate the area of a triangle given the Cartesian coordinates of the 3 vertices.

**Tetrahedral Volume**We can use matrices to calculate the volume of a tetrahedron given the Cartesian coordinates of the 4 vertices.

**Summary**Grab a PDF copy of the summary below.