# Find the perimeter of the triangle.[Graph]

Question

Find the perimeter of the triangle.

2021-03-06
The perimeter of a triangle is the sum of its side lengths so we first need to find the three side lengths of the triangle.
Since the side from (−3,−1) to (2,−1) is a horizontal segment, we can count the number of units between the two points to find its length. Since 2−(−3)=5, then the length of this side is 5.
Since the side from (2,−1) to (2,3) is a vertical segment, we can count the number of units between the two points to find its length. Since 3−(−1)=4, then the length of this side is 4.
The side between from (-3,-1) to (2,3) is not a horizontal or vertical segment so we need to use the distance formula $$\displaystyle{d}=\sqrt{{{\left({x}{2}-{x}{1}\right)}^{{{2}}}+{\left({y}{2}-{y}{1}\right)}^{{{2}}}}}$$ to find the lenght of this side:
$$\displaystyle{d}=\sqrt{{{\left(-{3}-{2}\right)}^{{{2}}}+{\left(-{1}-{3}\right)}^{{{2}}}}}$$ \ Substitute.
$$\displaystyle=\sqrt{{{\left(-{5}\right)}^{{{2}}}+{\left(-{4}\right)}^{{{2}}}}}$$ \ Substract.
$$\displaystyle=\sqrt{{{\left({25}+{16}\right)}}}$$ \ Evaluate the powers.
$$\displaystyle=\sqrt{{{41}}}$$ \ Add.
The length of the third side is then $$\displaystyle√{41}≈{6.4}.$$
The perimeter of the triangle is then about 5+4+6.4=15.4. The exact perimeter is $$\displaystyle{5}+{4}+\sqrt{{{41}}}={9}+\sqrt{{{41}}}$$

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