Geometric definition of the shortest distance from a point to a line

📚 Quick Study Guide

• 📏 The shortest distance from a point to a line is the length of the perpendicular segment from the point to the line.
• 📐 This perpendicular segment forms a right angle ($90^{\circ}$) with the line.
• ➗ To find this distance, you can use the formula: $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$, where $(x_0, y_0)$ is the point and $Ax + By + C = 0$ is the equation of the line.
• ✍️ Alternatively, find the equation of the perpendicular line passing through the point, find the intersection point of the two lines, and then calculate the distance between the original point and the intersection point.
• 💡 A common mistake is to use any arbitrary line segment from the point to the line, which will usually be longer than the perpendicular distance.

Practice Quiz

1. What geometric property defines the shortest distance from a point to a line?

   1. A) The angle bisector from the point to the line
   2. B) The perpendicular segment from the point to the line
   3. C) The parallel segment from the point to the line
   4. D) Any line segment from the point to the line

2. If a line is given by the equation $3x + 4y + 5 = 0$, what part of the distance formula represents the denominator?

   1. A) $|Ax_0 + By_0 + C|$
   2. B) $\sqrt{A^2 + B^2}$
   3. C) $A^2 + B^2$
   4. D) $Ax_0 + By_0 + C$

3. What angle does the shortest distance line segment form with the given line?

   1. A) $45^{\circ}$
   2. B) $60^{\circ}$
   3. C) $90^{\circ}$
   4. D) $0^{\circ}$

4. Given a point (1, 2) and a line $x + y = 0$, which method is best to find the shortest distance?

   1. A) Measure the distance with a ruler on a graph
   2. B) Use the distance formula: $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$
   3. C) Draw any line from the point to the line and measure its length
   4. D) Calculate the parallel distance

5. What does $(x_0, y_0)$ represent in the shortest distance formula $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$?

   1. A) A point on the line
   2. B) The origin (0, 0)
   3. C) The given point from which the distance is calculated
   4. D) The intersection of two lines

6. If the shortest distance from a point to a line is zero, what does this imply?

   1. A) The line is parallel to the x-axis
   2. B) The point lies on the line
   3. C) The line is parallel to the y-axis
   4. D) The point is at the origin

7. What is the shortest distance from the point (0, 0) to the line $x + y - 2 = 0$?

   1. A) $\sqrt{2}$
   2. B) 2
   3. C) $2\sqrt{2}$
   4. D) $\frac{\sqrt{2}}{2}$