Answer:
Explanation:
To find the length of each side of the triangle and verify if it is a right triangle, we can use the distance formula.
The distance formula calculates the distance between two points in a coordinate plane using the coordinates (x1, y1) and (x2, y2). It can be written as:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Now, let's calculate the lengths of the sides:
1. Side AB:
Using points A (-5, 1) and B (6, 0), we substitute the coordinates into the distance formula:
dAB = √((6 - (-5))^2 + (0 - 1)^2)
= √(11^2 + (-1)^2)
= √(121 + 1)
= √122
2. Side AC:
Using points A (-5, 1) and C (1, 6), we substitute the coordinates into the distance formula:
dAC = √((1 - (-5))^2 + (6 - 1)^2)
= √(6^2 + 5^2)
= √(36 + 25)
= √61
3. Side BC:
Using points B (6, 0) and C (1, 6), we substitute the coordinates into the distance formula:
dBC = √((1 - 6)^2 + (6 - 0)^2)
= √((-5)^2 + 6^2)
= √(25 + 36)
= √61
To verify if the triangle is a right triangle, we can check if the square of the longest side is equal to the sum of the squares of the other two sides (according to the Pythagorean theorem).
In this case, the longest side is BC, with a length of √61. The other two sides are AB (√122) and AC (√61).
According to the Pythagorean theorem, if BC^2 = AB^2 + AC^2, then the triangle is a right triangle.
Substituting the lengths:
(√61)^2 = (√122)^2 + (√61)^2
61 = 122 + 61
Since 61 is not equal to 183, the triangle is not a right triangle.
Therefore, the lengths of the sides are √122, √61, and √61, and the triangle is not a right triangle.