Given:
The figure of a triangle RST on a coordinate plane.
To find:
The perimeter of the triangle RST.
Solution:
From the given graph it is clear that the vertices of the given triangle are R(-2,7), S(-5,1) and T(-7,-5).
Distance formula:
[tex]D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
Using the distance formula, we get
[tex]RS=\sqrt{\left(-5-\left(-2\right)\right)^2+\left(1-7\right)^2}[/tex]
[tex]RS=\sqrt{\left(-5+2\right)^2+\left(1-7\right)^2}[/tex]
[tex]RS=\sqrt{\left(-3\right)^2+\left(-6\right)^2}[/tex]
on further simplification, we get
[tex]RS=\sqrt{9+36}[/tex]
[tex]RS=\sqrt{45}[/tex]
[tex]RS=3\sqrt{5}[/tex]
Similarly,
[tex]ST=\sqrt{\left(-7-\left(-5\right)\right)^2+\left(-5-1\right)^2}[/tex]
[tex]ST=2\sqrt{10}[/tex]
[tex]RT=\sqrt{\left(-7-\left(-2\right)\right)^2+\left(-5-7\right)^2}[/tex]
[tex]RT=13[/tex]
Now, the perimeter of the triangle RST is
[tex]Perimeter=RS+ST+RT[/tex]
[tex]Perimeter=3\sqrt{5}+2\sqrt{10}+13[/tex]
[tex]Perimeter=26.032759[/tex]
[tex]Perimeter\approx 26.0[/tex]
Therefore, the perimeter of the triangle RST is 26.0 units.
