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 xy-plane, we can use the formula for the distance between two points, considering one of those points to be (x,y) = (0,0). We square the x-value of the point that doesn't lie at the origin, then square the y-value of that point, then add the two squared values, and finally take the square root of the sum of the squares. In this case, our point P has the coordinates (x,y) = (-5,2). Therefore, x = -5 and y = 2. Squaring these values, we get

x2 = 25

and

y2 = 4

Adding the two squares yields

x2 + y2 = 25 + 4
= 29

Finally, we take the square root. The distance between point P and the origin (0,0) equals the square root of 29 units. The answer is D.

2. As we did when we worked out the solution to Question 1, we square the x-value of the point that doesn't lie at the origin, then square the y-value of that point, then add the two squared values, and finally take the square root of the sum of the squares. This time, our point Q has the coordinates (x,y) = (5,-5). Therefore, x = 5 and y = -5. Squaring these values, we get

x2 = 25

and

y2 = 25

Adding the two squares yields

x2 + y2 = 25 + 25
= 50

The distance between point Q and the origin equals the square root of 50 units. The correct choice is C.

3. When we want to find the distance between two points in the Cartesian xy-plane, we must square the difference between the x-values, then square the difference between the y-values, then add the squares, and finally take the square root of the sum of the squares. Let's give the coordinates of point P the name (x0,y0) and give the coordinates of point Q the name (x1,y1 ). We can calculate the distance d between the points P and Q using the formula

d = [(x1 x0)2 + (y1 y0)2]1/2

In this particular situation, we have

P = (x0,y0)
= (-5,2)

and

Q = (x1,y1)
= (5,-5)

Taking the differences in the x- and y-values, we get

x1 - x0 = 5 - (-5)
= 5 + 5
= 10

and

y1 - y0 = -5 - 2
= -7

Now our formula tells us that

d = [102 + (-7)2]1/2
= (100 + 49)1/2
= 1491/2

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 xy-plane, we should divide the difference in the y-values by the difference in the x-values, both taken in the same order. Usually, we'll take the values for the second point (in this case Q) and subtract the values for the first point (in this case P). The process goes as follows.

  • 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 P and Q in Figure 6-16. The answer is B.

5. In the Cartesian xy-plane, the point-slope equation for a line having slope m, passing through any particular point (x0,y0), has the form

y - y0 = m (x - x0)

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 P are (2,-5), we can verify that choice A really does represent the correct response. We can say that m = -7/10, x0 = 2, and y0 = -5 in the above generalized point-slope equation. Then, inputting these values, we get

y - 2 = (-7/10) [x - (-5)]

which simplifies to

y - 2 = (-7/10) (x + 5)

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 xy-plane. It's

y = ax2 + bx + c

where a, b, and c represent constants, and a doesn't equal 0. In this situation, our equation is

y = -2x2 + 8x - 3

so a = -2, b = 8, and c = -3. Now let's remember the fact that the equation for the x-coordinate of the vertex point of a parabola in the above form (call it x0) is

x0 = -b / (2a)

When we input the known values for a and b, we get

x0 = -8 / [2 x (-2)]
= -8 / (-4)
= 2

We can calculate the y-value of the vertex point (call it y0) by inputting x0 = 2 into the parabola's equation and then working through the arithmetic. When we carry out that task, we get

y0 = -2 x 22 + 8 x 2 - 3
= -8 + 16 - 3
= 5

We have determined the parabola's vertex point coordinates as

(x0,y0) = (2,5)

The correct choice is B.

7. Let's review the general equation for a circle in the Cartesian xy-plane. If (x0,y0) are the coordinates of the circle's center and the radius equals r units, then we can describe the circle as

(x - x0)2 + (y - y0)2 = r2

When we input the values x0 = -2, y0 = 7, and r = 14 to the above equation, we get

[x - (-2)]2 + (y - 7)2 = 142

which simplifies to

(x + 2)2 + (y - 7)2 = 196

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.