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.) I , r/y

Verified step by step guidance

1. Identify the quadrant given in the problem, which is Quadrant I. In this quadrant, both x and y coordinates are positive, so \(x > 0\) and \(y > 0\).
2. Recall the formula for \(r\), the distance from the origin to the point \((x, y)\): \(r = \sqrt{x^2 + y^2}\). Since \(x^2\) and \(y^2\) are always non-negative, \(r\) is always positive.
3. Analyze the ratio given: \(\frac{r}{y}\). Since \(r > 0\) and \(y > 0\) in Quadrant I, both numerator and denominator are positive.
4. Because both numerator and denominator are positive, the ratio \(\frac{r}{y}\) must be positive.
5. To confirm your understanding, sketch the coordinate plane, plot a point in Quadrant I, and visually verify that \(r\) and \(y\) are positive, reinforcing why the ratio is positive.

Key Concepts

Coordinate Plane Quadrants

The coordinate plane is divided into four quadrants, each with specific signs for x and y coordinates. In Quadrant I, both x and y are positive, which affects the sign of ratios involving these values.

Distance from Origin (r)

The distance r from the origin to a point (x, y) is given by r = √(x² + y²), which is always positive. This value represents the radius in polar coordinates and is crucial for understanding ratios involving r.

Sign of Ratios Involving Coordinates

The sign of a ratio like r/y depends on the signs of numerator and denominator. Since r is always positive, the sign of r/y depends solely on y's sign, which varies by quadrant, helping determine if the ratio is positive or negative.