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2D Coordinate Geometry

In the 1600s, René Descartes married algebra and geometry to create the Cartesian plane.

Problem Solving

         

I left my house and walked 800 meters north, before I reached the traffic intersection.
I then turned right, and walked a further 1500 meters east, to reach the amusement park.

What is the straight-line distance (in meters) from my house to the amusement park?

\(a, b, c \) are real constants such that all the lines represented by \(ax+by+c=0\) are concurrent at a point \(P\). This point \(P\) lies on the line \(7a+2b+8c=0\). What are the coordinates of \(P\)?

Find the equation of the line which is tangent to the circle \[ (x-3)^2 + (y-11)^2 = 53 \] at the point \(P = ( 5, 4). \)

If four points on the \(xy\)-plane \( A=(-1, a)\), \(B=(b, 0) \), \(C=(15, 1)\), and \(D=(2, 21) \) are the vertices of parallelogram \( ABCD \), what is the value of \( a + b \)?

Parallelogram \(ABCD\) has three vertices \[ A=(28, 24),\ B=(3,4) \textrm{ and }\ C=(14, 36). \] If the intersection point of the two diagonals \(AC\) and \(BD\) is \( M=(a,\ b) \), what is the value of \( a + b \)?

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