Suppose that the point (x,y) is in the indicated quadrant. Decide whether the given is positive or negative.
25.) II, x/r
26.) III, y/r
27.) IV, x/y
r is always positive.
you should know the values of x and y in each of the quadrants.
It is ESSENTIAL that you learn the CAST rule.
To determine whether the given quantities are positive or negative based on the indicated quadrant, we can use the following guidelines:
- In Quadrant I, both x and y coordinates are positive.
- In Quadrant II, the x coordinate is negative and the y coordinate is positive.
- In Quadrant III, both x and y coordinates are negative.
- In Quadrant IV, the x coordinate is positive and the y coordinate is negative.
Based on these guidelines, here are the answers for each given quantity:
25.) In Quadrant II: x is negative. Therefore, x/r is negative.
26.) In Quadrant III: y is negative. Therefore, y/r is negative.
27.) In Quadrant IV: x is positive and y is negative. Therefore, x/y is positive.
To determine whether a given value is positive or negative in a particular quadrant, we need to understand the signs of the coordinates in that quadrant.
In the Cartesian coordinate system, there are four quadrants:
- Quadrant I: Positive x, and positive y values
- Quadrant II: Negative x, and positive y values
- Quadrant III: Negative x, and negative y values
- Quadrant IV: Positive x, and negative y values
Let's apply this knowledge to answer each question:
25.) II, x/r
In Quadrant II, the x-coordinate is negative, while the y-coordinate is positive. Now, we need to determine whether x/r (x divided by r) is positive or negative. To find the sign of x/r, we need to know the sign of r.
If r is positive, then x/r would be negative in Quadrant II because of the negative x-coordinate.
If r is negative, then x/r would be positive in Quadrant II. This is because dividing a negative x-coordinate by a negative r value results in a positive ratio.
Therefore, the sign of x/r in Quadrant II depends on the sign of r.
26.) III, y/r
In Quadrant III, both the x-coordinate and y-coordinate are negative. Similarly, we need to determine the sign of y/r (y divided by r) based on the sign of r.
If r is positive, then y/r would be negative in Quadrant III because of the negative y-coordinate.
If r is negative, then y/r would be positive in Quadrant III. Dividing a negative y-coordinate by a negative r value yields a positive ratio.
Therefore, the sign of y/r in Quadrant III depends on the sign of r.
27.) IV, x/y
In Quadrant IV, the x-coordinate is positive, while the y-coordinate is negative. Now, let's determine the sign of x/y (x divided by y) based on the signs of x and y.
Since we divide a positive x-coordinate by a negative y-coordinate, x/y would be negative in Quadrant IV.
Thus, the sign of x/y in Quadrant IV would be negative.
In summary:
- In Quadrant II, the sign of x/r depends on the sign of r.
- In Quadrant III, the sign of y/r depends on the sign of r.
- In Quadrant IV, the sign of x/y is negative.