Solved examples: classifying rectangles and squares using slope

📚 Quick Study Guide

• 📐 Definition: A rectangle is a quadrilateral with four right angles. A square is a special type of rectangle where all sides are equal.
• 📏 Slopes:
  • Parallel lines have equal slopes.
  • Perpendicular lines have slopes that are negative reciprocals of each other (i.e., if one slope is $m$, the other is $-\frac{1}{m}$).
• ➕ Conditions for Rectangles:
  • Opposite sides have equal slopes (parallel).
  • Adjacent sides have slopes that are negative reciprocals of each other (perpendicular).
• ➕ Conditions for Squares:
  • All sides are equal in length, *and*
  • Adjacent sides are perpendicular.
• 💡 Tip: Calculate the slopes of all four sides. Check for parallel and perpendicular relationships to determine if the quadrilateral is a rectangle or a square. Also, check side lengths to distinguish a rectangle from a square.

Practice Quiz

1. What condition must be met for two lines to be considered perpendicular?
   1. A) Their slopes are equal.
   2. B) Their slopes are reciprocals.
   3. C) Their slopes are negative reciprocals.
   4. D) Their slopes are equal to 1.
2. A quadrilateral has vertices A(1, 2), B(4, 2), C(4, 5), and D(1, 5). What are the slopes of sides AB and BC?
   1. A) AB: 0, BC: undefined
   2. B) AB: undefined, BC: 0
   3. C) AB: 1, BC: -1
   4. D) AB: 2, BC: -2
3. If line segment PQ has a slope of 2, what is the slope of a line perpendicular to PQ?
   1. A) 2
   2. B) -2
   3. C) $\frac{1}{2}$
   4. D) $-\frac{1}{2}$
4. Which property distinguishes a square from a rectangle?
   1. A) All angles are right angles.
   2. B) Opposite sides are parallel.
   3. C) All sides are equal.
   4. D) Opposite sides are equal.
5. A quadrilateral has vertices E(0, 0), F(3, 0), G(3, 3), and H(0, 3). What is the shape of the quadrilateral?
   1. A) Rectangle
   2. B) Parallelogram
   3. C) Square
   4. D) Trapezoid
6. The slopes of two adjacent sides of a quadrilateral are 3 and $-\frac{1}{3}$. What can you conclude about these sides?
   1. A) They are parallel.
   2. B) They are perpendicular.
   3. C) They are equal in length.
   4. D) They form an acute angle.
7. Which of the following sets of slopes could represent the sides of a rectangle?
   1. A) 2, 2, -$\frac{1}{2}$, -$\frac{1}{2}$
   2. B) 2, -2, 2, -2
   3. C) 2, $\frac{1}{2}$, 2, $\frac{1}{2}$
   4. D) 2, -$\frac{1}{2}$, 2, $\frac{1}{2}$