To find the perimeter of a triangle, we need to calculate the sum of the lengths of all three sides.
To calculate the length of a side, we can use the distance formula, which is based on the Pythagorean theorem. The distance formula states that the distance between two points in a coordinate plane, (x₁, y₁) and (x₂, y₂), is given by:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
Using this formula, we can find the lengths of the three sides of the triangle.
Let's label the given points:
A = (1, -4)
B = (-7, 0)
C = (-4, -3)
To find the length of side AB, we substitute the coordinates of points A and B into the distance formula:
d_AB = √((-7 - 1)² + (0 - (-4))²)
= √((-8)² + 4²)
= √(64 + 16)
= √80
= 4√5
Similarly, we can find the lengths of sides BC and AC.
d_BC = √((-4 - (-7))² + (-3 - 0)²)
= √(3² + (-3)²)
= √(9 + 9)
= √18
= 3√2
d_AC = √((1 - (-4))² + (-4 - (-3))²)
= √(5² + (-1)²)
= √(25 + 1)
= √26
Now, we can calculate the perimeter by summing up the lengths of all three sides:
Perimeter = d_AB + d_BC + d_AC
= 4√5 + 3√2 + √26
Therefore, the exact answer for the perimeter of the triangle with vertices (1, -4), (-7, 0), and (-4, -3) is 4√5 + 3√2 + √26.