Nathan draws a triangle on the coordinate grid. He marks his points at (1,5)
, (4,1)
, and (−4,−5)
. What is the perimeter of the triangle, rounded to the nearest tenth of a unit? Explain the steps that Nathan must take to find the solution.
To find the perimeter of a triangle, Nathan must first calculate the distance between each pair of points.
Using the distance formula, which states that the distance between two points (x₁, y₁) and (x₂, y₂) is given by:
d = √[(x₂ - x₁)² + (y₂ - y₁)²],
Nathan can find the distances between the points:
d₁ = √[(4 - 1)² + (1 - 5)²] = √[3² + (-4)²] = √(9 + 16) = √25 = 5 units,
d₂ = √[(-4 - 1)² + (-5 - 1)²] = √[(-5)² + (-6)²] = √(25 + 36) = √61 ≈ 7.81 units,
d₃ = √[(-4 - 1)² + (-5 - 5)²] = √[(-5)² + (-10)²] = √(25 + 100) = √125 ≈ 11.18 units.
Next, Nathan must add up the distances to find the perimeter:
Perimeter = d₁ + d₂ + d₃ = 5 + 7.81 + 11.18 = 23.99 ≈ 24 units.
Therefore, the perimeter of the triangle, rounded to the nearest tenth of a unit, is 24 units.