This document outlines a lesson focused on the distance formula derived from the Pythagorean theorem, aiming to help learners accurately derive the formula and solve related problems. It includes practical applications, examples, and exercises for determining the distance between two points in a Cartesian plane. Key concepts emphasize the relationship between the Pythagorean theorem and the distance formula, providing step-by-step solutions to reinforce understanding.

1. Lesson 11.1
The Distance Formula

2. Learning Competency
At the end of the lesson, the learners should be able
to derive the distance formula.

3. Objectives
At the end of this lesson, the learners should be able to
do the following:
● Accurately derive the distance formula.
● Correctly solve for the distance between two points
using the distance formula.
● Correctly solve word problems involving distance
formula.

4. One of the practical applications of the Pythagorean Theorem is
to determine the distance between two points in a Cartesian
plane. This is done by comparing their vertical and horizontal
distances.

5. This is useful in several real-life applications, such as
determining the distance between two locations on a map,
which is used in the navigation apps on your phone.

6. In this lesson, you will learn how to use the distance formula to
determine the distance between two points.

7. Essential Questions
● How is Pythagorean Theorem related to the distance
formula?
● In what other areas of Mathematics can the distance
formula be applied?

8. Learn about It!
This theorem states that in any given right triangle, the square
of the length of the hypotenuse is equal to the sum of the
squares of the lengths of the other two sides.
In a right triangle Δ𝐴𝐵𝐶, it the measures of the legs are 𝑎 and
𝑏, and the measure of the hypotenuse is 𝑐, it follows that
𝑎2
+ 𝑏2
= 𝑐2
.
Pythagorean Theorem

9. Learn about It!
Example:
To find the hypotenuse of a right triangle whose legs measure
3 m and 4 m, we follow these steps.
Let 𝑎 = 3 and 𝑏 = 4. Substitute the given to the formula
𝑎2
+ 𝑏2
= 𝑐2
.
Pythagorean Theorem

10. Learn about It!
Example:
32
+ 42
= 𝑐2
9 + 16 = 𝑐2
25 = 𝑐2
25 = 𝑐2
5 = 𝑐
𝑐 = 5
Pythagorean Theorem

11. Learn about It!
Example:
Thus, the hypotenuse of the given right triangle is 𝟓 𝐦.
Pythagorean Theorem

12. Learn about It!
This formula is used to find the distance between two distinct
points on the Cartesian plane.
Given two points 𝑥1, 𝑦1 and 𝑥2, 𝑦2 , the distance between the
two points can be measured as
𝑑 = 𝑥2 − 𝑥1
2 + 𝑦2 − 𝑦1
2
Distance Formula

13. Learn about It!
Example:
To find the distance between points 2, 3 and −3, −1 , we
follow these steps:
Let 𝑥1 = 2, 𝑥2 = −3, 𝑦1 = 3, and 𝑦2 = −1. Substitute the given to
the distance formula 𝑑 = 𝑥2 − 𝑥1
2 + 𝑦2 − 𝑦1
2.
Distance Formula

14. Learn about It!
Example:
𝑑 = −3 − 2 2 + −1 − 3 2
𝑑 = −5 2 + −4 2
𝑑 = 25 + 16
𝑑 = 41
Distance Formula

15. Learn about It!
Example:
Hence, the distance between points 2, 3 and −3, −1 is 𝟒𝟏
units.
Distance Formula

16. Try it!
Let’s Practice
Example 1: Find the length of the hypotenuse of the right
triangle in the figure given below.
𝑐 = ?
𝑎 = 6
𝑏 = 10

17. Solution to Let’s Practice
Solution:
Substitute the values to the Pythagorean equation.
Example 1: Find the length of the hypotenuse of the right
triangle in the figure given below.

18. Solution to Let’s Practice
Solution:
𝑎2
+ 𝑏2
= 𝑐2
6 2
+ 10 2
= 𝑐2
36 + 100 = 𝑐2
136 = 𝑐2
Example 1: Find the length of the hypotenuse of the right
triangle in the figure given below.

19. Solution to Let’s Practice
Solution:
136 = 𝑐2
2 34 = 𝑐
𝑐 = 2 34
Therefore, the length of the hypotenuse is 𝟐 𝟑𝟒 units.
Example 1: Find the length of the hypotenuse of the right
triangle in the figure given below.

20. Try it!
Let’s Practice
Example 2: Miguel’s house is mapped on the coordinate grid
with the city hall as the origin. His home is located at point
(2, 4) and his best friend’s home is at point (−4, 12). If the scale
used in the grid is 1 unit ∶ 1 km, how far are their homes from
each other?

21. Solution to Let’s Practice
Solution:
From the given, let 𝑥1 = 2, 𝑥2= −4, 𝑦1 = 4, and 𝑦2 = 12.
Substitute these values to the distance formula.
Example 2: Miguel’s house is mapped on the coordinate grid with the city hall as the origin.
His home is located at point (2, 4) and his best friend’s home is at point (−4, 12). If the scale
used in the grid is 1 unit ∶ 1 km, how far are their homes from each other?

22. Solution to Let’s Practice
Solution:
𝑑 = 𝑥2 − 𝑥1
2 + 𝑦2 − 𝑦1
2
= −4 − 2 2 + 12 − 4 2
= −6 2 + 8 2
= 36 + 64
= 100
= 10
Example 2: Miguel’s house is mapped on the coordinate grid with the city hall as the origin.
His home is located at point (2, 4) and his best friend’s home is at point (−4, 12). If the scale
used in the grid is 1 unit ∶ 1 km, how far are their homes from each other?

23. Solution to Let’s Practice
Solution:
Therefore, the distance between Miguel’s home and his best
friend’s home is 10 km.
Example 2: Miguel’s house is mapped on the coordinate grid with the city hall as the origin.
His home is located at point (2, 4) and his best friend’s home is at point (−4, 12). If the scale
used in the grid is 1 unit ∶ 1 km, how far are their homes from each other?

24. Try It!
Individual Practice:
1. Find the distance between the points (−3, −4) and (2, 6).
2. Given the points 𝑎, 6 and 28, 13 , find the value of 𝑎 so
that the distance between the two given points is 25
units.

25. Try It!
Group Practice: To be done in two to five groups
Two cars departed from Manila at the same time. Car A
traveled 5 km north and 3 km west while Car B traveled 2
km south and 4 km east. How far are the two cars from
one another?

26. Key Points
• The Pythagorean Theorem states that in any given right
triangle, the square of the length of the hypotenuse is
equal to the sum of the squares of the lengths of the
other two sides
o In a right triangle, the length of the legs are 𝑎 and 𝑏,
and the hypotenuse is 𝑐. It follows that 𝑎2
+ 𝑏2
= 𝑐2
.

27. Key Points
• The distance formula is used to find the distance
between two distinct points on the Cartesian plane.
o Given two points 𝑥1, 𝑦1 and 𝑥2, 𝑦2 , the distance
between the two points can be measured as
𝑑 = 𝑥2 − 𝑥1
2 + 𝑦2 − 𝑦1
2.

28. Bibliography
Pierce, Rod. “Distance Between 2 Points.” Math Is Fun. Retrieved 6 June 2019 from
https://www.mathsisfun.com/algebra/distance-2-points.html
Stapel, Elizabeth. “The Distance Formula.” Purple Math. Retrieved 17 May 2019 from
http://bit.ly/2JM6gtW