📂Geometry

Area of a Triangle Enclosed by a Straight Line and the x and y Axes

Overview

Questions asking whether it’s possible to find the maximum or minimum value, or the tangent line often involve calculating the area of such triangles $S$. Of course, finding the area of a triangle is not difficult, but it would be even better if one could remember a simple formula and solve it immediately.

Theorem

• The $y$ intercept of the line $y=mx+n$ is $n$, and the $x$ intercept is $-\frac { n }{ m }$. The area of the triangle enclosed by this line and the $x$ axis, $y$ axis is as follows: $$S = \left| \frac { n^{ 2 } }{ 2m } \right|$$
• The $y$ intercept of the line $ax+by+c=0$ is $-\frac { c }{ b }$, and the $x$ intercept is $-\frac { c }{ a }$. The area of the triangle enclosed by this line and the $x$ axis, $y$ axis is as follows: $$S = \left| \frac { { c }^{ 2 } }{ 2ab } \right|$$