Rectangular Geometry

Trivium

  1. Given an origin point and two perpendicular coordinate axes in a plane, use a ruler to estimate the rectangular coordinates of any point in that plane.
  2. Given the rectangular coordinates of two points in space, calculate the distance between them and their midpoint.
  3. Given the rectangular coordinates of two points in space, determine an equation of the line that contains those points.
  4. Given a point in space and a slope, determine an equation of the line with that slope passing through that point, and vice versa: given an linear equation, recognize it as the equation of a line and determine its slope.
  5. Given a point in space and a radius, determine the equation of the circle with that radius centered at that point, and vice versa: given an equation corresponding to a circle, recognize it as the equation of a circle and determine its center and radius.

Problems

  1. Consider the points \((1,1)\) and \((3,-3).\) Determine a point on the \(y\)-axis that is equidistant to those points.
  2. Consider a line segment with one endpoint at the coordinates \((4,1)\) and midpoint at the coordinates \((-3,4).\) What are the coordinates of the other endpoint?
  3. A football field is 160′ wide and 360′ long (remember the end zones). Suppose you have to walk from one corner of the football field to the opposite corner. How far is this? As a percentage, how much shorter is it to cut across the field diagonally rather than walk around the outside?
  4. What is an equation for the circle whose diameter is a line segment with endpoints at the coordinates \((1,5)\) and \((4,9)?\)
  5. What is an equation for the circle that passes through the points \((0,0)\) and \((-24,0)\) and \((0,70)?\) What are the area and circumference of this circle? What are the \(y\)-coordinates of the points on this circle that have an \(x\)-coordinate of \(23?\)
  6. Consider the circle with equation \(x^2 + y^2 = 16.\) There are two points on this circle with an \(x\)-coordinate of one. What are the \(y\)-coordinates of these points?
  7. Adeline and Beatrix are testing out a new pair of walkie-talkies (two-way radios) in residential downtown Grand Junction. After turning their radios to the same channel, they decide that Adeline will stay in place at an intersection of two roads while Beatrix jogs around. The packaging says that the walkie-talkies have an effective range of one mile. The streets downtown run east-west and north-south and the blocks are about \(\frac{1}{12}\) of a mile long.
    1. Suppose that Beatrix jogs eight blocks west and nine blocks north. Are Adeline and Beatrix still within communication range?
    2. After meeting back up with Adeline, suppose now that Beatrix jogs eleven blocks south and then starts jogging east. How many blocks can she jog before losing contact with Adeline?
    3. After meeting back up with Adeline, suppose now that Beatrix plans on jogging to her favorite downtown coffee shop. She enters the address of the coffee shop into her GPS to determine the quickest route, and the GPS reports it’ll be a 1.2 mile jog (along streets) to get there. Does this necessarily mean the coffee shop it outside of communication range with Adeline?
  8. Consider the three points in the \(xy\)-plane with coordinates \((1,3)\) and \((-2,8)\) and \((4,7).\) Suppose that \(P\) is another point such that these four points altogether are the vertices of a parallelogram. What are all the possibilities for the coordinates of \(P?\)
  9. A rhombus is a four-sided shape characterized by the fact that both pairs of its opposite sides are parallel. If the points \((3,6)\) and \((-1,2)\) are the coordinates of opposite vertices of a rhombus in the plane, what’s an equation for the line that passes through the other two vertices?
  10. Without appealing to technology, explain, as if convincing a skeptical peer in the class, how you can tell that the three points \((-1,3)\) and \((1,-1)\) and \((5,-9)\) all lie on the same line.
  11. Given a line segment, its perpendicular bisector is the unique line that is perpendicular to the segment and passes though its midpoint. Determine an equation for the perpendicular bisector of the line segment with endpoints \((-1,-4)\) and \((5,8).\)
  12. What is the area of the triangle in the \(xy\)-plane formed by the \(x\)-axis, the \(y\)-axis, and the line given by the equation \(y = 5-2x?\) In general, any line that is not vertical or horizontal will form a triangle with the \(x\)- and \(y\)-axis. Given such a line \(y = mx + b\) devise a formula for the area of this triangle in terms of \(m\) and \(b.\)
  13. What is an equation for the line that passes through the point \((22,-6)\) and whose \(x\)-intercept is five more than its \(y\)-intercept?
  14. Suppose a line intersects the \(x\)-axis at the number \(12,\) and the area of the triangle formed by this line and the positive \(x\)- and \(y\)-axis is \(30.\) What must the slope of this line be?
  15. While uncommonly used in curricular education, the equation \[ \frac{x}{a}+\frac{y}{b} = 1\] may very well serve as a template for an equation of a line in place of the ubiquitous “slope-intercept form”. What are geometric interpretations of the values of \(a\) and \(b\) in this template?
  16. A line is tangent to a circle at a point if it intersects the circle at only that point; it must touch the circle but not pass into its interior. Verify that the point \((12,35)\) lies on the circle \(x^2+y^2 = 1369,\) and determine an equation of the line tangent to the circle at that point.
  17. A rain gutter should be installed at a slight incline, usually called the pitch of the gutter. But “pitch” is just another word for “grade” or “slope”. A common recommendation is that the gutter should drop by ¼″ for every 10′ of roofing.
    1. What slope does this recommendation correspond to?
    2. If you have to install a gutter along a 55′ length of roofing, how much higher does one end of the gutter be over the other?
    3. If you have to install a gutter along a 55′ length of roofing, what length of gutter do you need?
  18. You may have seen road signs like the one here that indicate the road is declined steeply at a specific grade. For example, an 8% grade (decline) indicates that there is an 8% loss in elevation per horizontal distance travelled — e.g. for every 100′ a truck travels horizontally it’ll lose 8′ of elevation. This corresponds directly with the “slope” of the road. Considering the surface of the road as a line, the slope of that line will be -8%.
    1. Suppose a truck drives one mile horizontally on a 6% downhill grade. How many feet of elevation did it lose? (Remember 1 mile is 5280 feet.)
    2. Suppose a truck drives on a 6% downhill grade and loses one mile of elevation. How far did it travel horizontally?
    3. Suppose a truck drives across one mile of road on a 6% downhill grade — i.e. the truck’s odometer indicates it’s travelled one mile. How much elevation did it lose?
  19. What is the closest the line \(y=3x+5\) comes to the origin? In general, how close does the line with equation \(y = mx+b\) come to the origin?
  20. The equation \(x^2+y^2-2x+6y+3=0\) corresponds to a circle. Without referring to technology, determine the center and radius of this circle by algebraically re-writing in the form \((x-h)^2+(y-k)^2=r^2.\) Hint: if you don’t know where to start, refresh yourself on the algebraic technique called “completing the square”.
  21. How far apart are the parallel lines \(y = \frac{1}{2}x + 13\) and \(y = \frac{1}{2}x - 4?\) In general, for two parallel lines given by equations \(y = mx + b_1\) and \(y = mx + b_2,\) devise a formula in terms of \(m\) and \(b_1\) and \(b_1\) for the distance between them.
  22. For any three non-colinear points in space, there is a unique circle that passes through those points. For such points \((x_1,y_1)\) and \((x_2, y_2)\) and \((x_3, y_3)\) devise an equation for the unique circle containing those points.