Answer:
[tex]DT=\sqrt{169+36}=\sqrt{205}[/tex]
Step-by-step 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 [tex](c=\sqrt{a^{2}+b^{2}})[/tex]
- 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
∴ [tex]DT=\sqrt{(DN)^{2}+(TN)^{2}}[/tex]
∴ [tex]DT=\sqrt{(13)^{2}+(6)^{2}}[/tex]
∴ [tex]DT=\sqrt{169+36}=\sqrt{205}[/tex]
