Shortest Distance Between a
Point and a Line
Quarter 4- Module 1
Infinite lines can be drawn from a point to a line. The shortest
distance is the one drawn perpendicular to the line. In the figure
below, the shortest distance from 𝑃1 to the line y = mx + b is
through 𝑃2 .
𝑷𝟏
A
𝑷𝟐
B
To find the shortest distance from a
point to a line:
𝒅=
𝒚𝟏 − 𝒎𝒙𝟏 − 𝒃
𝒎𝟐 + 𝟏
EXAMPLE 1. Find the distance from (3, 1) to y = 2x+4.
Given:
Solution:
𝒅=
𝒅=
𝒅=
𝒅=
𝒅=
𝑥1 = 3
𝒚𝟏 − 𝒎𝒙𝟏 − 𝒃
𝒎𝟐
𝑦1 = 1
m=2
b=4
𝒅=
+𝟏
𝟏 − 𝟐(𝟑) − 𝟒
𝟐𝟐 + 𝟏
𝒅=
𝟗
𝟓
•
𝟓
𝟓
𝟗 𝟓
𝟓
𝟏−𝟔−𝟒
𝟒+𝟏
−𝟗
𝟓
𝟗
𝟓
Therefore, the distance from (3, 1)to y = 2x + 4 is
𝟗 𝟓
𝟓
units.
EXAMPLE 2. What is the shortest distance from y = -x +3 to A(-3, 1).
Given:
𝑥1 = −3
𝑦1 = 1
m = -1
b=3
Solution:
𝒅=
𝒅=
𝒅=
𝒅=
𝒅=
𝒚𝟏 − 𝒎𝒙𝟏 − 𝒃
𝒎𝟐
+𝟏
𝟏 − (−𝟏)(−𝟑) − 𝟑
(−𝟏)𝟐 +𝟏
𝟏−𝟑−𝟑
𝟏+𝟏
−𝟓
𝟐
𝟓
𝟐
𝒅=
𝟓
𝟐
•
𝟐
𝟐
𝟓 𝟐
𝒅=
𝟐
Therefore, the distance from y = -x +3 to A(-3, 1)
is
𝟓 𝟐
𝟐
units.
When the general equation of the line, Ax + By + C = 0 is
given, then use this alternative formula,
𝑨𝒙𝟏 + 𝑩𝒚𝟏 + 𝑪
𝒅=
𝑨𝟐 + 𝑩𝟐
EXAMPLE 3. Find the shortest distance from (1, -2) to 3x – 4y + 4 = 0.
Given:
𝑦1 = −2
𝑥1 = 1
A=3
B = -4
Solution:
𝒅=
𝒅=
𝒅=
𝑨𝒙𝟏 + 𝑩𝒚𝟏 + 𝑪
𝑨𝟐
+
𝒅=
𝑩𝟐
𝟑(𝟏) + (−𝟒)(−𝟐) + 𝟒
𝟑𝟐 + (−𝟒)𝟐
𝟑+𝟖 +𝟒
𝟗 + 𝟏𝟔
𝒅=
C=4
𝟏𝟓
𝟐𝟓
𝟏𝟓
𝟓
𝒅=𝟑
Therefore, the distance from (1, -2) to 3x – 4y + 4
= 0 is 3 units.
EXAMPLE 4. How far is the distance from (-1, 2) to y = 4x – 5?
Either formula can be used to find the required distance.
Solution:
Method 1: Using 𝒅 =
Given:
𝒅=
𝒅=
𝒅=
𝒅=
𝒅=
𝒚𝟏 − 𝒎𝒙𝟏 − 𝒃
𝒎𝟐
+𝟏
𝟐 − 𝟒(−𝟏) − (−𝟓)
𝟒𝟐 + 𝟏
m=4
b = -5
y = 4x – 5
-4x + y + 5 = 0
-1(-4x + y + 5 = 0)
4x – y – 5 = 0
Given:
𝟐+𝟒+𝟓
𝟏𝟔 + 𝟏
𝒅=
𝟏𝟏
𝒅=
𝟏𝟕
𝟏𝟏
𝟏𝟕
•
𝟏𝟏 𝟏𝟕
𝒅=
𝟏𝟕
𝑨𝒙𝟏 +𝑩𝒚𝟏 +𝑪
𝒎𝟐 +𝟏
𝑦1 = 2
𝑥1 = −1
Method 1: Using 𝒅 =
𝒚𝟏 −𝒎𝒙𝟏 −𝒃
𝟏𝟕
𝟏𝟕
𝒅=
𝒅=
𝒅=
𝑨𝟐 +𝑩𝟐
Rewrite the equation in the general
form.
𝑦1 = 2
𝑥1 = −1
A=4
𝑨𝒙𝟏 + 𝑩𝒚𝟏 + 𝑪
B = -1 C = -5
𝑨𝟐 + 𝑩𝟐
𝟒(−𝟏) + (−𝟏)(𝟐) + (−𝟓)
𝟒𝟐
+
(−𝟏)𝟐
𝟏𝟏
𝒅=
𝒅=
𝟏𝟕
𝟏𝟕
•
𝟏𝟕
𝟏𝟏 𝟏𝟕
𝟏𝟕
−𝟒 − 𝟐 − 𝟓
𝟏𝟔 + 𝟏
−𝟏𝟏
𝟏𝟕
𝟏𝟏
𝟏𝟕
Thus, the distance from (-1, 2) to y = 4x
– 5 is
𝟏𝟏 𝟏𝟕
𝟏𝟕
units.
QUIZ 1
MODULE 1
Shortest Distance Between a Point and a Line
Find the distance between the given points and lines.
1. (1, 2); 3x + 4y – 1 = 0
𝒅=
𝑨𝒙𝟏 + 𝑩𝒚𝟏 + 𝑪
𝑨𝟐 + 𝑩𝟐
2. (-5, 1); y = 2x + 0
𝒅=
𝒚𝟏 − 𝒎𝒙𝟏 − 𝒃
𝒎𝟐 + 𝟏