Geometry
# 2D Coordinate Geometry

Circle $O$ is centered at the point of origin with point $P = (3 , 4)$ lying on it. The red line $l : 3x + 4y - 7 = 0$ intersects the circle at points $A$ and $B,$ as shown.

What is the area of quadrilateral $AOBP?$

$\large {\left\{\begin{matrix}A_{n+1}=\alpha A_n+(1-\alpha)B_n \\ B_{n+1}=\alpha B_n+(1-\alpha)C_n \\ C_{n+1}=\alpha C_n+(1-\alpha)A_n\end{matrix}\right.}$

Let $A_1,B_1,C_1$ be three distinct non-collinear points on the coordinate plane.

Also $A_n,B_n,C_n$ satisfy the recurrence relation above ($0<\alpha<1$).

Then $\displaystyle\lim_{n\to\infty}A_n$ is the $\text{\_\_\_\_\_\_\_\_\_\_}$ of the triangle $A_1B_1C_1$.

Did you know that we can have coordinate systems where the coordinate axes are not perpendicular to each other? Such coordinate systems are known as *oblique coordinate systems*.

The above figure shows an oblique coordinate system. In such systems, the $x$-coordinate of a point is found by measuring the distance from the point to the $y$-axis parallel to the $x$-axis. Similarly, the $y$-coordinate of a point is found by measuring the distance from the point to the $x$-axis parallel to the $y$-axis. The red point in the above figure has the coordinates $(a, b)$.

**Question:** We are given two points: $A \ (1,6)$ and $B \ (5,2)$ in an oblique coordinate system where the angle between the positive axes is $60^{\circ}$. What is the distance between the two points, $AB$?

The $\triangle AOB$ consists of point $A = (0 , 1)$, point $O = (0 , 0)$, and point $B$ lying somewhere on the $x$-axis.

Let $P$ be the point in the first quadrant such that $AP = PB$ and $AO \parallel PB$, as shown above.

If the length of $OP$ is $\sqrt{194}$, what is the area of the quadrilateral $AOBP?$

$(3,5)$ on the plane. It goes 10 units in the direction of positive $x$-axis. Then it turns towards right making an angle of 90 degrees with its initial direction and goes ahead 5 units, Now it again takes a right turn and goes ahead 2.5 units. It keeps turning right and travels half the distance traveled in the turn just before it . This goes on without end. Find the final distance of the particle from the origin $(0,0)$?

A particle is on the pointThe answer would come in the form square root of $x$, submit your answer as $x$.