Interactive Quiz: Derive the Distance Formula From Two Points on a Coordinate Plane

📚 Quick Study Guide

• 📍 Coordinate Plane: A 2D plane formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Points are located using ordered pairs $(x, y)$.
• 📐 Distance Formula: Derived from the Pythagorean theorem, it calculates the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a coordinate plane.
• ➗ The Formula: The distance, $d$, is given by the formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• 💡 Steps to Apply:
  1. Label the coordinates: $(x_1, y_1)$ and $(x_2, y_2)$.
  2. Substitute the values into the distance formula.
  3. Simplify the expression inside the square root.
  4. Calculate the square root to find the distance.
• ➕ Important Note: The order of subtraction ($x_2 - x_1$ or $x_1 - x_2$) doesn't matter because you're squaring the result. Same goes for $y_2 - y_1$.

✍️ Practice Quiz

1. What is the distance formula used to calculate?
   1. The area of a triangle.
   2. The slope of a line.
   3. The distance between two points on a coordinate plane.
   4. The volume of a sphere.
2. Given the points (1, 2) and (4, 6), which of the following represents the correct substitution into the distance formula?
   1. $\sqrt{(1-4)^2 + (2-6)^2}$
   2. $\sqrt{(1+4)^2 + (2+6)^2}$
   3. $\sqrt{(4+1)^2 + (6+2)^2}$
   4. $\sqrt{(4-2)^2 + (6-1)^2}$
3. What is the distance between the points (0, 0) and (3, 4)?
   1. 5
   2. 7
   3. 25
   4. 12
4. If the distance between two points is $\sqrt{25}$, what is the simplified value of the distance?
   1. 25
   2. 6.25
   3. 5
   4. $\sqrt{5}$
5. The distance formula is derived from which theorem?
   1. The Angle Bisector Theorem
   2. The Pythagorean Theorem
   3. The Intersecting Chords Theorem
   4. The Law of Sines
6. Calculate the distance between (-2, 3) and (1, -1).
   1. $\sqrt{5}$
   2. 5
   3. $\sqrt{7}$
   4. $\sqrt{25}$
7. What is the distance between the points (5, -2) and (5, 4)?
   1. 0
   2. 6
   3. 10
   4. 12