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/y

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Recall that the point (x, y) lies in Quadrant IV. In this quadrant, the x-coordinate is positive and the y-coordinate is negative.

The ratio given is \( \frac{x}{y} \). Since \( x > 0 \) and \( y < 0 \) in Quadrant IV, the numerator is positive and the denominator is negative.

A positive number divided by a negative number results in a negative value.

Therefore, the ratio \( \frac{x}{y} \) in Quadrant IV is negative.

To visualize this, sketch the coordinate plane, mark Quadrant IV, and plot a point with positive x and negative y to see why the ratio is negative.

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Coordinate Plane Quadrants

The coordinate plane is divided into four quadrants, each with specific sign conventions for x and y coordinates. In Quadrant IV, x is positive and y is negative. Understanding these sign rules is essential for determining the sign of ratios involving x and y.

Sign of Ratios in Different Quadrants

The sign of a ratio like x/y depends on the signs of numerator and denominator. Since x and y have known signs in each quadrant, the ratio's sign can be deduced by dividing their signs. For example, in Quadrant IV, x/y is positive divided by negative, resulting in a negative ratio.

Distance from Origin (r = √(x² + y²))

The distance r from the origin to the point (x, y) is always positive and is calculated using the Pythagorean theorem. While r itself is positive, it helps relate x and y to trigonometric functions and confirms the point's position relative to the origin.