Triangle ABC has vertices A(0,4),B(1,2),and C(4,6). Determine whether triangle ABC is a right triangle.Explain.
How do I do this?
take the slopes of lines AB, AC, and BC.
Are any of those 3 slopes opposite reciprocals of each other?
To determine whether triangle ABC is a right triangle, you can use the Pythagorean theorem or check if the slopes of any two sides are negative reciprocals of each other.
1. Using the Pythagorean theorem:
Start by calculating the lengths of each side of the triangle using the distance formula:
AB = √((x₂ - x₁)² + (y₂ - y₁)²)
BC = √((x₃ - x₂)² + (y₃ - y₂)²)
AC = √((x₃ - x₁)² + (y₃ - y₁)²)
In this case, we have:
AB = √((1-0)² + (2-4)²) = √(1 + 4) = √5
BC = √((4-1)² + (6-2)²) = √(9 + 16) = √25 = 5
AC = √((4-0)² + (6-4)²) = √(16 + 4) = √20 = 2√5
Now, check if the square of the length of the longest side (BC) is equal to the sum of the squares of the other two sides (AB and AC).
BC² = AB² + AC²
25 = 5 + (2√5)²
25 = 5 + 4(5)
25 = 5 + 20
25 = 25
Since the equation holds true, we can conclude that triangle ABC is a right triangle.
Alternatively,
2. Checking the slopes:
Calculate the slopes of the two sides AB and AC using the formula:
Slope = (y₂ - y₁)/(x₂ - x₁)
For AB:
Slope_AB = (2-4)/(1-0) = -2/1 = -2
For AC:
Slope_AC = (6-4)/(4-0) = 2/4 = 1/2
If the slopes of two sides are negative reciprocals of each other, then the triangle is a right triangle. In this case, -2 and 1/2 are not negative reciprocals, so the triangle is not a right triangle.
Therefore, using either method, we can conclude that triangle ABC is not a right triangle.