How to find the area of a triangle after a linear transformation

Expert intro: A linear map changes areas by the absolute value of its determinant. So you do not need to redraw the transformed triangle point by point unless the question asks for coordinates as well.

Detailed walkthrough

The original triangle is $OA B$ with

$$A=(0,2),\qquad B=(3,0).$$

Its area is

$$\text{Area}(OAB)=\frac12\left|\det\begin{pmatrix}0&3\\2&0\end{pmatrix}\right| =\frac12| -6|=3.$$

The matrix of the transformation is

$$M=\begin{pmatrix}3&2\\5&8\end{pmatrix}.$$

A linear transformation multiplies area by $|\det M|$.

Compute the determinant:

$$\det M=3\cdot 8-2\cdot 5=24-10=14.$$

So the area of triangle $OA'B'$ is

$$14\times 3=42.$$

Therefore,

$$\boxed{42}.$$

💡 Pitfall guide

Do not try to transform the area by adding or averaging coordinates. Area scaling under a linear map comes from the determinant, not from the entries themselves. Also, remember the absolute value of the determinant, since area cannot be negative.

🔄 Real-world variant

If the triangle did not include the origin, you could still find the area after transformation by first computing the original area from coordinates, then multiplying by $|\det M|$. If the transformation were not linear but affine, you would need to account for translation separately, though translation still does not change area.

🔍 Related terms

determinant, area scaling, linear transformation