Triangle RST with coordinates R(2, 3), S(4, 4), and T(5, 0) is not a right triangle.
What is distance formula?
A distance formula, as its name suggests, gives the distance (the length of the line segment). For example, the distance between two points is the length of the line segment connecting them.
Formula of distance between two points formula:
\(d = \sqrt{(x2-x1)^{2} + (y2-y1)^{2} }\)
Where ,
( x1,y1 ) and (x2,y2) are points
According to the question
Given :
Triangle RST with coordinates R(2, 3), S(4, 4), and T(5, 0) .
To prove:
Triangle RST is a right angled triangle
Proof:
(x1,y1) = R(2, 3)
(x2,y2) = S(4, 4)
(x3,y3) = T(5, 0)
now, applying distance formula to find distance between RS
\(d = \sqrt{(x2-x1)^{2} + (y2-y1)^{2} }\)
substituting the value
\(d = \sqrt{(4-2)^{2} + (4-3)^{2} }\)
\(d = \sqrt{(2)^{2} + (1)^{2} }\)
d = \(\sqrt{5}\)
Now, distance between ST
(x2,y2) = S(4, 4)
(x3,y3) = T(5, 0)
substituting the value
\(d = \sqrt{(5-4)^{2} + (0-4)^{2} }\)
\(d = \sqrt{(1)^{2} + (4)^{2} }\)
\(d = \sqrt{17}\)
Now, distance between TR
(x3,y3) = T(5, 0)
(x1,y1) = R(2, 3)
\(d = \sqrt{(5-2)^{2} + (3-0)^{2} }\)
\(d = \sqrt{(3)^{2} + (3)^{2} }\)
\(d = 3\sqrt{2}\)
Applying Pythagoras theorem
As right angle triangle always follow Pythagoras theorem .
\(x^{2} + y^{2} = z^{2}\)
5 + 17 ≠ 18
Hence, Triangle RST with coordinates R(2, 3), S(4, 4), and T(5, 0) is not a right triangle.
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