Find the measure of the sides abc with vertices a(1 ,3) b(5,-2) c(0,-4) classify the triangle by its sides

ab = √41  units, ac = 2√5 units and bc = √29 units.

Since all its sides are not equal. So, it is a scalene triangle.

Step-by-step explanation:

Since we have the vertices at a(1 ,3) b(5,-2) c(0,-4), the length of each side is given by √[(x₂ - x₁)² + (y₂ - y₁)²] where x₁ and x₂ are the corresponding x-coordinates and y₁ and y₂ are the corresponding y-coordinates.

So for the side ab with vertices a(1 ,3) b(5,-2), x₁ = 1, x₂ = 5, y₁ = 3 and y₂ = -2. Substituting these into the equation, we have

ab = √[(x₂ - x₁)² + (y₂ - y₁)²] = √[(5 - 1)² + (-2 - 3)²] = √[4² + (-5)²] = √[16 + 25] = √41  units

So for the side ac with vertices a(1 ,3) c(0,-4), x₁ = 1, x₂ = 0, y₁ = 3 and y₂ = -4. Substituting these into the equation, we have

ab = √[(x₂ - x₁)² + (y₂ - y₁)²] = √[(0 - 1)² + (-4 - 3)²] = √[(-1)² + (-7)²] = √[1 + 49] = √50 = 2√5 units

So for the side bc with vertices b(5 ,-2) c(0,-4), x₁ = 5, x₂ = 0, y₁ = -2 and y₂ = -4. Substituting these into the equation, we have

ab = √[(x₂ - x₁)² + (y₂ - y₁)²] = √[(0 - 5)² + (-4 - (-2))²] = √[(-5)² + (-4 + 2)²] = √[(-5)² + (-2)²] = √[25 + 4] = √29 units

Since ab = √41  units, ac = 2√5 units and bc = √29 units, all its sides are not equal. So, it is a scalene triangle.

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