The Distance Formula, Slides of Mathematics

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Determine the distance between two points of objects in a Cartesian plane; Derive the distance formula; Find the distance between two points using the distance formula; and

Objective

s

Appreciate how the distance formula facilitates in finding solutions to real-life problems.

Cartesian Coordinate Plane II I III IV

Plot the following ordered pairs in the Cartesian Plane and identify where quadrant it belongs. 1. (4, 3)

**2. (-2, -5)

3. ( -6, 1)
4. (1, -3)
5. (2, 4)**

Plot the following ordered pairs in the Cartesian Plane and identify where quadrant it belongs.

Quadrant I

Plot the following ordered pairs in the Cartesian Plane and identify where quadrant it belongs.

1)Quadrant II

Plot the following ordered pairs in the Cartesian Plane and identify where quadrant it belongs.

Quadrant IV

4 units

5 units

Along Visayas Avenue are the houses of two nursing graduates, namely Ana and Carlo. Because of the rising cases of COVID-19 in Quezon City, they decided to apply as volunteers at the HOPE facility that is located at QCMC. The location of their houses is illustrated on the coordinate plane as shown. a. What is the distance between Ana's house and Carlo's house? How about the distance between HOPE and Carlo's house? Ana’s HOPE Carlo’s

4 units

5 units

Ana’s (^) Carlo’s b. Use line segments to connect the points representing the house of Ana (A), HOPE (B), and the house of Carlo (C). What kind of triangle is formed?? c. Give the coordinates of points A, B, and C. A(1, 1) d. Using the coordinates, how do you determine the horizontal distance from A to C or C to A? the vertical distance from B to C or C to B?

B(5, 6) C(5, 1)

Horizontal distance from A to C or C to A |1-5|= 4 |6-1|= 5 Vertical distance from B to C or C to B |1-6|= 5 HOPE A

B

C

Replace the coordinates of A by (, ) and B by (, ). The coordinates of C would be (, ). Horizontal distance from A to C or C to A Vertical distance from B to C or C to B

or

or

A

B (, ) C

(, ) (, ) To compute for the distance from A to B we will use the Pythagorean Theorem and substitute the values. 𝑨𝑩 𝟐 =( 𝒙 𝟐 − 𝒙 𝟏 ) 𝟐 +( 𝒚 𝟐 − 𝒚 𝟏 ) 𝟐 𝑨𝑩 = √ ( 𝒙 𝟐 − 𝒙 𝟏 ) 𝟐 +( 𝒚 𝟐 − 𝒚 𝟏 ) 𝟐 𝑨𝑩 | 𝟐 =| 𝑨𝑪 | 𝟐 +| 𝑩𝑪 | 𝟐

Distance Formula

AB

A

B

(, ) C

(, ) (, ) is an equation that can be used to find the distance between any pair of points on the coordinate plane. The Distance Formula 𝒅 = √ ( 𝒙 𝟐 − 𝒙 𝟏 ) 𝟐 +( 𝒚 𝟐 − 𝒚 𝟏 ) 𝟐 𝒅 = √ ( 𝒙 𝟐 − 𝒙 𝟏 ) 𝟐 + ( 𝒚 (^) 𝟐^ − 𝒚 (^) 𝟏 ) 𝟐

The TLE Department organized a tree planting project for SHS students of ERHS. Tom and Jerry participated in the said activity and it served as their project. Let us find the distance between the trees they have planted. https://theconversation.com/social-distancing-how-we-overcome- fear-of-one-another-to-embrace-a-new-normal- In the time of the Covid-19 pandemic, after an initial lockdown period, the government implement social distancing. Retail spaces have been transformed to safely accommodate customers. With the use of signages and messages, we are now learning to adapt to social distancing in all areas of life. In this illustration, let us find the distance between the two spaces. Do you think social distancing would help prevent COVID-19 from spreading? Example 1

Due to the calamity brought by typhoon Odette, The Philippine National Red Cross in Manila would like to send their relief goods to the affected province of Tacloban, Leyte. Let us calculate the distance between Manila and Tacloban, Leyte. https://www.worldmapsonline.com/philippines-political- educational-map-from-academia-maps/ Example 2