๐ Topic Summary
Coordinate geometry provides a powerful way to verify the properties of quadrilaterals using algebraic techniques. By plotting the vertices of a quadrilateral on a coordinate plane, we can calculate side lengths using the distance formula and determine if sides are parallel or perpendicular by examining their slopes. These calculations allow us to rigorously prove whether a quadrilateral is a parallelogram, rectangle, square, rhombus, or trapezoid.
The distance formula, $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, helps calculate the lengths of the sides. The slope formula, $m = \frac{y_2 - y_1}{x_2 - x_1}$, helps determine if lines are parallel (same slope) or perpendicular (slopes are negative reciprocals of each other).
๐ Part A: Vocabulary
Match the term with its correct definition:
- Quadrilateral
- Parallelogram
- Rectangle
- Rhombus
- Trapezoid
- A quadrilateral with one pair of parallel sides.
- A parallelogram with four right angles.
- A quadrilateral with four sides.
- A parallelogram with four congruent sides.
- A quadrilateral with two pairs of parallel sides.
โ๏ธ Part B: Fill in the Blanks
Use the words distance, slope, parallel, perpendicular, and coordinate to complete the paragraph.
In ______ geometry, we can verify properties of quadrilaterals. To find side lengths, we use the ______ formula. To determine if sides are ______ or ______ , we use the ______ formula.
๐ค Part C: Critical Thinking
Explain how you would use coordinate geometry to prove that a quadrilateral with vertices A(1, 1), B(5, 1), C(5, 4), and D(1, 4) is a square.