Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Find the area of the triangle in the plane $R^2$ bounded by the lines $y = x$, $y = -3x+8$, and $3y + 5x = 0$

I know that I can find the area of the triangle by taking the half of the area of the parallelogram the points make. But I don't know how to convert those equations to points so I can take the vectors and calculate it's determinant.

share|cite|improve this question
up vote 5 down vote accepted

Find the points $(x_i, y_i)$ at which each pair of equations intersect. There are three such pairs, so three points of intersection, which will be the vertices of the triangle. From the vertices $(x_i, y_i)$, you can determine the two vectors you need, which you can use as the columns of the matrix for which the absolute value of the determinant, multiplied by 1/2, will give you area.

Given $\;y = x$, $y = -3x+8$, and $3y + 5x = 0$


Vertex 1: At what point does $y = x$ and $y = -3x + 8$ intersect? When $x = -3x + 8$. Solving for $x$, gives us $x = 2$, which in this case, will also equal $y$.

So vertex 1 is $(2, 2)$. Proceed in a similar manner to determine:

Vertex 2: where $y = x$ and $3y + 5x = 0$.

Vertex 3: where $y - -3x + 8$ and $3y + 5x = 0$

share|cite|improve this answer
Thank you. A very clear explanation. I solve the problem, the area is 16 – Randolf Rincón Fadul Apr 10 '13 at 21:06
You're very welcome! – amWhy Apr 10 '13 at 21:08

Or from the vertices, you can apply the formula:

$$Area = \dfrac12 \left|\begin{array}{cccc} x_1 && x_2 && x_3 && x_1 \\ y_1 && y_2 && y_3 && y_1 \end{array}\right|$$

If you know how to use it.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.