Question

plot the points D (-6,-2) and T (7,-8). Describe how to find the distance between D and T using the Pythagorean Theorem

Answer 1

`DT=√(169+36)=√(205)`

Explanation:

* Lets explain how to solve the problem

- The coordinates of point D are (-6 , -2)

- The coordinates of point T are (7 , -8)

- We want to find the distance between D and T using Pythagorean

Theorem

- Pythagorean Theorem is:

in any right angle triangle the length of the hypotenuse (opposite

side to the right angle) = square root of the sum of the squares of

the other two sides
`(c=\sqrt{a^(2)+b^(2)})`

- So to make the right angle with points D and T, we must to draw a

horizontal segment from point D and vertical segment from point T

cross each other at point N whose coordinates are (x , y)

- Look to the attached graph for more understand

- D is the red point , N is the green point and T is the blue point

* Lets find x and y

- The y-coordinates of all points lie on a horizontal segment are equal

∵ DN is a horizontal segment

∴ y-coordinate of point D = y-coordinate of point N

∵ y-coordinate of point D is -2

∵ y- coordinate of point N = -2

∴ y = -2

- The x-coordinates of all points lie on a vertical segment are equal

∵ TN is a vertical segment

∴ x-coordinate of point T = x-coordinate of point N

∵ x-coordinate of point T is 7

∵ x- coordinate of point N = 7

∴ x = 7

∴ The coordinates of point N are (7 , -2)

* Lets find the length of DN and TN

∵ The length of the horizontal segment is the difference between

the x-coordinates of its endpoints

∵ x-coordinate of D is -6 and x-coordinate of N is 7

∴ DN = 7 - (-6) = 7 + 6 = 13

∵ The length of the vertical segment is the difference between

the y-coordinates of its endpoints

∵ y-coordinate of T is -8 and y-coordinate of N is -2

∴ TN = (-2) - (-8) = -2 + 8 = 6

* Lets use Pythagorean Theorem to find DT

- In Δ DNT

∵ m∠N = 90°

∵ DT is opposite to ∠N

∴
`DT=\sqrt{(DN)^(2)+(TN)^(2)}`

∴
`DT=\sqrt{(13)^(2)+(6)^(2)}`

∴
`DT=√(169+36)=√(205)`