Question

quadrilateral qrst on a coordinate plane. she uses the slopes of the segments t phrases to complete the steps kristie uses to prove that qrst is a rectangle. e slope formula to show that both pairs of opposite sides have slopes that are equ that at least one pair of adjacent sides have slopes that are reciprocals.

Explanation:

To determine the correct terms for proving \( QRST \) is a rectangle:

Step 1: Opposite Sides of a Rectangle

A rectangle is a parallelogram with right angles. For a quadrilateral to be a parallelogram, both pairs of opposite sides must be parallel. Parallel lines have equal slopes (since slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \), and equal slopes imply the same steepness and direction). Thus, the first blank (for opposite sides) should be "equal" (to show parallelism).

Step 2: Adjacent Sides of a Rectangle

For a parallelogram to be a rectangle, adjacent sides must be perpendicular (form right angles). Two lines are perpendicular if the product of their slopes is \( -1 \), which means one slope is the negative reciprocal of the other (e.g., if slope \( m_1 = 2 \), then \( m_2 = -\frac{1}{2} \), so \( m_1 \cdot m_2 = -1 \)). The term "reciprocals" alone is incorrect here—they must be negative reciprocals. However, if the context assumes the "negative" is implied (or the question has a typo), the key idea is that perpendicular lines have slopes that are negative reciprocals. For the purpose of this problem, the second blank (for adjacent sides) should reflect perpendicularity, so the correct term (accounting for the problem’s phrasing) is that adjacent sides have slopes that are negative reciprocals (though the problem uses "reciprocals"—likely a simplification, but the core is perpendicularity via slope relationship).

Answer:

• First blank: equal (to show opposite sides are parallel, as parallel lines have equal slopes).
• Second blank: negative reciprocals (to show adjacent sides are perpendicular, as perpendicular lines have slopes that are negative reciprocals; if the problem uses "reciprocals" as a simplification, it still implies perpendicularity).

(Note: If the problem strictly uses "reciprocals" as given, the intended answer is likely "negative reciprocals" for perpendicularity, but the first blank is "equal" for parallel opposite sides.)