Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) III , r/y

Verified step by step guidance

1. Recall that the point (x, y) lies in Quadrant III, where both x and y coordinates are negative. This means x < 0 and y < 0 in this quadrant.
2. Remember the definition of r, which is the distance from the origin to the point (x, y), given by the formula \(r = \sqrt{x^2 + y^2}\). Since it is a distance, r is always positive regardless of the quadrant.
3. The ratio given is \(\frac{r}{y}\). Since r is positive and y is negative in Quadrant III, the numerator is positive and the denominator is negative.
4. When dividing a positive number by a negative number, the result is negative. Therefore, the ratio \(\frac{r}{y}\) will be negative in Quadrant III.
5. To confirm your understanding, sketch the coordinate plane, plot a point in Quadrant III, and visually verify the signs of x, y, and r, then consider the ratio \(\frac{r}{y}\).

Key Concepts

Coordinate Plane Quadrants

The coordinate plane is divided into four quadrants, each with specific signs for x and y coordinates. In Quadrant III, both x and y values are negative. Understanding the sign of coordinates in each quadrant helps determine the sign of expressions involving x and y.

Distance Formula and Radius r

The radius r is defined as r = √(x² + y²), representing the distance from the origin to the point (x, y). Since it is a distance, r is always positive regardless of the quadrant. This property is crucial when analyzing ratios involving r.

Sign of Ratios Involving Coordinates

When evaluating the sign of a ratio like r/y, consider the signs of numerator and denominator separately. Since r is positive and y is negative in Quadrant III, the ratio r/y will be negative. This approach helps determine the positivity or negativity of trigonometric ratios.