Geometry Demystified, 2nd edition |

Stan Gibilisco |

Explanations for Quiz Answers in Chapter 6 |

1. When we want to find the distance of a point from the origin in the
Cartesian
and
Adding the two squares yields
Finally, we take the square root. The distance between point |

2. As we did when we worked out the solution to Question 1, we
square the
and
Adding the two squares yields
The distance between point |

3. When we want to find the distance between two
points in the Cartesian
In this particular situation, we have
and
Taking the differences in the
and
Now our formula tells us that
or the square root of 149 units. The answer is C. |

4. When we want to find the slope of a line that passes through two
points with known coordinates in the Cartesian - The
*y*-values are -5 for point*Q*and 2 for point*P*, so the difference is -5 - 2 = -7. - The
*x*-values are 5 for point*Q*and -5 for point*P*, so the difference is 5 - (-5) = 5 + 5 = 10. - The difference in the
*y*-values divided by the difference in the*x*-values (taken in the same order) equals -7/10.
The ratio -7/10 represents the slope of the line passing
through points |

5. In the Cartesian
We've determined that the slope of the line is -7/10. By
elimination, we can deduce that the correct answer to this
problem must be A, because it's the only choice where the slope
obviously has the correct value! However, when we note that the
coordinates of point
which simplifies to
That equation exactly matches the one in choice A. |

6. Let's remember the general form of the equation for a parabola
that appears in the Cartesian
where
so
When we input the known values for
We can calculate the
We have determined the parabola's vertex point coordinates as ( The correct choice is B. |

7. Let's review the general equation for a circle in the Cartesian
( When we input the values [ which simplifies to ( The circle's center has the coordinates (-2,7). The answer is C. |

8. When we worked out the solution to Question 7, we got the general
equation for the circle to
yield the correct specific result when r = 14. The circle
therefore has a radius of 14 units. The answer is D. |

9. Suppose that we see two curves, lines, or
geometric figures in the Cartesian xy-plane, and we know (or
can determine) the equations that the figures represent. If we solve those
two equations simultaneously, then the number of real-number solutions to
those equations will always equal the number of points at which the
geometric figures intersect. In this case, line A and curve B don't
intersect anywhere, so the pair of equations has no real-number solutions.
The correct choice is A. |

10. According to the above-described methodology, we can have reasonable confidence that the equations for curves B and C have one simultaneous real-number solution because those curves appear to intersect at a single point. The answer is B. |