# Thread: proof: triangle of variable length on the Cartesian plane

1. ## proof: triangle of variable length on the Cartesian plane

A variable triangle OAB is formed by a straight line passing through the point P(a, b) on the Cartesian plane and cutting through the x-axis and y-axis at A and B respectively.
If the angle OAB=theta, find the area of triangle OAB in terms of a, b, and theta.

2. Originally Posted by shawli
A variable triangle OAB is formed by a straight line passing through the point P(a, b) on the Cartesian plane and cutting through the x-axis and y-axis at A and B respectively.
If the angle OAB=theta, find the area of triangle OAB in terms of a, b, and theta.
Hello Shawli:

grad$\displaystyle =tan(\theta)$

$\displaystyle tan(\theta)=\frac{b-B}{a}$

$\displaystyle b-a.tan(\theta)=B$

$\displaystyle tan(\theta)=\frac{b}{a-A}$

$\displaystyle \frac{a.tan(\theta)-b}{tan(\theta)}=A$

$\displaystyle Area=\frac{1}{2}.A.B$ (Substitute the values of A & B)

Hope this helps

3. Name the points $\displaystyle B0,c)~\&~Ad,0)$.
Then the area of the triangle is $\displaystyle \frac{cd}{2}$.
The slope of the line is $\displaystyle -\tan(\theta)$.
Use that to find $\displaystyle c~\&~d$.

### a variable triangle OAB is formed by a straight line passing through the point(a,b) on the cartesian plane and cutting the x-axis and y- axis at the A and B respectively. if angle OAB = m , find the area of triangle OAB in terms of a,b and m

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