Coordinates

Author:: Adam

Problem 1

Which of the following points have coordinates with the same sign? (To illustrate: The coordinates of (1,1) have the same sign because they're both positive. The coordinates of (1,0) do not have the same sign because one is positive and the other is not.)

Problem 2.

Which of the points above satisfies the inequation x > 0? (I.e. which of them has a positive x-coordinate?) Which of them satisfies y = 0?

Problem 3.

Which of the points in problem 1 satisfy the equation xy = 1?

Problem 4.

Find the distance of every point in Problem 1 from the origin.

Problem 5.

Find two points (whether labeled or not) which are equally distant from A and F in problem 1. Find two labeled points which are equally distant from B and D.

Problem 6.

Let A(1,2) and B(4,0) be two points and call AB the line-segment joining them. Find the midpoint of AB and then find the point which is 1/3 of the distance from A to B.

Problem 7.

Let A(1,1), B(1,-1), C(-1,-1), and D(-1,1) be points in the plane. Find the average of all of their x-coordinates, and the average of all of their y-coordinates. Call these values

. Plot the points A, B, C, D, and

. What do you notice? Suppose you draw the triangle with vertices A(1,1), B(-1,1), and C(0,-1). Again plot A, B, C, and

. What do you notice?

Problem 8.

Consider the line that runs through points A(0,4) and B(2,0). One of the points on this line has as its x-coordinate 1. Which point is it? Which point has as its x-coordinate -1?

Problem 9.

Here we generalize what we did in problem 8. In the diagram below points A(0,4) and B(2,0) are marked and so is the line through them. A random point, labeled C, with x-coordinate equal to x is also marked. Use similar triangles to determine the y coordinate in terms of x. Hint: You know the legs of the smaller triangle are lengths 4 and 2 obviously, and by similar triangles the proportions must be equal with the large triangle. Also, whatever x and y are, we know that the legs of the large triangle have lengths 4-y and x. Set up the equality of the ratios and solve for y.

Problem 10.

The above example shows that every point on the line satisfies the equation y = 4-2x. We often write this as y = -2x + 4 and say that this is in slope-intercept form because the -2 is the slope and the 4 is the intercept. Draw the line with slope 2 and intercept -1.

Problem 11.

What are the slope and intercept of the line satisfying the equation y = x?

Problem 12.

As we've seen a line is the same thing as a set of points satisfying an equation of the form y = mx + b where m is the slope and b the intercept. We can find a similar definition of a circle in terms of an equation. If a circle is centered at the point (h,k) and has radius r, what equation must it satisfy? The labeled edges in the diagram below are a hint.

Problem 13.

Draw all points that satisfy the equation

. Find the equation of the circle with center at (2,-2) and radius 1. Draw all points that satisfy the equation

Problem 14.

A point is called a lattice point if its coordinates are integers. For instance, in the following diagram, only points A and B are lattice points. Find all of the lattice points in the circle centered at the origin with radius 5. Hint: Draw the circle on grid paper with good accuracy to get good guesses. Make sure that after you do this, you can prove that your guess is correct. This means being able to demonstrate that the point you've picked must really be on the circle--you need a mathematical demonstration, not just a diagram.