Orientation Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Vertices of triangle ABC are at A(0,0)A(0,0), B(4,0)B(4,0), C(2,3)C(2,3). Using the signed area formula, determine the orientation.

Solution

  1. 1
    Step 1: Signed area =12[xA(yB−yC)+xB(yC−yA)+xC(yA−yB)]= \frac{1}{2}[x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B)].
  2. 2
    Step 2: =12[0(0−3)+4(3−0)+2(0−0)]= \frac{1}{2}[0(0-3) + 4(3-0) + 2(0-0)].
  3. 3
    Step 3: =12[0+12+0]=6= \frac{1}{2}[0 + 12 + 0] = 6.
  4. 4
    Step 4: Positive signed area means counterclockwise (positive) orientation.

Answer

Counterclockwise orientation (positive).
The signed area (shoelace formula) gives a positive value for counterclockwise vertex ordering and a negative value for clockwise ordering. The absolute value gives the actual area. This is a powerful algebraic way to detect orientation without drawing the figure.

About Orientation

Orientation is the directional sense of a geometric figure — whether its vertices are ordered clockwise or counterclockwise. It describes how a shape is 'facing' in space, and is preserved by rotations and translations but reversed by reflections.

Learn more about Orientation →

More Orientation Examples

Example 1 easy

A triangle has vertices listed counterclockwise as A, B, C. After a reflection, the vertices appear

Example 3 easy

Does a 90° rotation change the orientation of a figure?

Example 4 hard

A left-handed glove is reflected in a mirror. Can the reflected image be rotated in 3D to match the

← All Orientation Examples Orientation Concept