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.) IV , x/r

Verified step by step guidance

1. Recall that the ratio given is \( \frac{x}{r} \), where \( r = \sqrt{x^2 + y^2} \) is the distance from the origin to the point \( (x, y) \). Since \( r \) is a square root of sums of squares, it is always positive.
2. Identify the quadrant of the point. The problem states the point is in Quadrant IV. In Quadrant IV, the \( x \)-coordinate is positive and the \( y \)-coordinate is negative.
3. Since \( x > 0 \) in Quadrant IV and \( r > 0 \) always, the ratio \( \frac{x}{r} \) is a positive number divided by a positive number.
4. Therefore, the ratio \( \frac{x}{r} \) must be positive in Quadrant IV.
5. To confirm, you can sketch the coordinate plane, plot a point in Quadrant IV, and visualize that \( x \) is positive and \( r \) is the hypotenuse (always positive), reinforcing the positivity of the ratio.

Key Concepts

Coordinate Plane Quadrants

The coordinate plane is divided into four quadrants, each with specific signs for x and y coordinates. In Quadrant IV, x is positive and y is negative. Understanding the sign of coordinates in each quadrant helps determine the sign of ratios involving x and y.

Distance from Origin (r)

The distance r from the origin to a point (x, y) is given by r = √(x² + y²). Since squares are always non-negative, r is always positive. This ensures that ratios involving r in the denominator maintain the sign of the numerator.

Sign of Ratios in Trigonometry

Ratios like x/r correspond to trigonometric functions (e.g., cosine θ). The sign of such ratios depends on the signs of numerator and denominator. Since r is positive, the sign of x/r depends solely on x, which is positive in Quadrant IV, making x/r positive.